00 - MTH 117 - Algebra (ASU)

Solve for x: (3x - 3)(x² + 10x + 21) = 0

(3x - 3)(x² + 10x + 21) = 0 (3x - 3)(x + 7)(x + 3) = 0 x = 1, -7, -3

Solve for w: 5w² - 15w = 0

5w² - 15w = 0 5w(w - 3) = 0 w = 3, 0

Mutliply: 81x + 36 4x - 16 -------- * ------- 8x - 32 9x + 4

9/2

If an equation ends up: 2y-2y = 4-4 0 = 0 What is the value of y?

All real numbers are solutions.

What is the quadratic formula and what is it used for?

Allows one to solve for x from: ax² + bx + c = 0

What is the different between semiannual, biannual, and biennial?

Biannual means twice per year, and semiannual means every six months (i.e. every half year) -- the same in the end but with a slight technical difference. Biennial means every two years (i.e. every other year).

How would you give the domain and range of a graph with several coordinates (see attached picture) using set notation? I.e. What would the format look like?

Domain: {-3, 1, 2} Range: {-1, 0}

Find the domain of the function and write in interval notation: f(x) = √(x) + 3

Find the set of all values that make √(x) a real number (i.e. when x is greater than or equal to zero). x >= 0 [0, ∞) * bracket -> includes point parenthesis -> point not included (can't include infinity)

What is the difference between even and odd functions?

Graphs: Even functions are are symmetric with respect to the y-axis. Odd functions are symmetric with respect to their origin. The only function that is both even and odd is f(x) = 0 Functions: If all exponents in the function are even, it's even. If all exponents are odd, it's odd. If there is a mix of even and odd exponents, it's neither. * Function s does not cross the y-axis evenly, and is not symmetric with respect to the origin (apparently??).

How would you find all the real zeros of: f(x) = -4x² - 6x + 1 On a graphing calculator?

Input the function. Click on "Zero". Find the two x-intercepts or a vertex point where it touches the x-intercept.

On a graph with multiple points and/or lines, how does one determine if it is a valid function?

It is a valid function if each x-value has only one y-value.

Subtract: d/3 - 4/7

LCD = 21 cross-multiply 7d - 12 ------- 21

Add: 2/7 + 5/7c

LCD = 7c 2c + 5 ------- 7c

Solve for y: √(y - 3) + 12 = 5

No solution

What is the solution to the following equations: -x + 4y = -4 x - 4y = 4

There are infinite solutions 0 = 0 y = (x-4)/4

Find the slopes of the lines passing through: a) (-6, 9) (-6, -7) b) (-9, -6) (2, -6)

a) the x coordinates are the same so the slope is undefined (i.e. vertical) - rise over run (-7 - 9) / (-6 - (-6)) = -16 / 0 b) the y coordinates are the same so the slope is 0 (-6 - (-6)) / (2 - (-9)) = 0 / 11

Write a⁵ * a² without exponents.

a*a*a*a*a*a*a

Which of these are polynomials: a(x) = x³ - 0.5x + 7x⁵ b(x) = -9 c(x) = 5x⁵ + 2√x + 3x d(x) = (2 - x²) / 3 e(x) = 8x² + 3x⁻⁴ f(x) = 7x⁴ + 4/x²

a, b, d * Coefficients must be real numbers ** Exponents must be whole numbers (including no negatives) *** Remember exponents in denominator (e.g. 4/x²) become negative

What is the greatest common factor of 13a³ and 7a⁴?

a³ 13, 7 == 1 aaa, aaaa == aaa

Simplify the following fraction taken to the 4th power. ( a⁴ ) ⁴ ( --- ) ( -3b² )

a¹⁶ ----- 81 b⁸

For each pair of function f and g, find f(g(x)) and g(f(x)). Are they inverses of each other? f(x) = 2/x g(x) = 2/x f(x) = x+3 g(x) = -x + 3

f(x) = 2/x & g(x) = 2/x f(g(x)) = x g(f(x)) = x Inverses of each other if both equal x f(x) = x+3 & g(x) = -x + 3 f(g(x)) = -x + 6 g(f(x)) = -x Not inverses

Give the domains of these equations in interval notation: g(x) = x / (x² + 16) f(x) = (x + 3) / x² - 9 h(x) = √(x-2)² -3

g(x) = x / (x² + 16) There are no values of x that would result in an asymptote or a hole. (-∞, ∞) f(x) = (x + 3) / x² - 9 v-asymptote = 3, -3 (-∞, -3)u(-3, 3)u(3, ∞) h(x) = √(x-2)² -3 = x - 5 [2, ∞)

Given g(x) = x² - 9x, find the equation of the secant line passing through (-1, g(-1)) and (8, g(8)). Write your answer in the form y = mx + b.

g(x) = x² - 9x (-1) = 1 + 9 = (-1, 10) (8) = 64 - 72 = (8, -8) Slope = (-8 -10) / (8 - (-1)) = -2 y = -2x + b 10 = 2 + b b = 8 y = -2x + 8

The area of a rectangle is 27 m² , and the length of the rectangle is 3 m less than twice the width. Find the dimensions of the rectangle.

w(2w - 3) = 27 2w² - 3w - 27 = 0 (2w - 9)(w + 3) = 0 w = -3, 4.5 = 4.5 m (can't be negative) L = 2w - 3 = 6 m

Convert the following radical expression to exponential: ⁵√13³

x ᵐ⁄ⁿ = ⁿ√xᵐ 13³⁄⁵

What is the meaning of a graph being symmetric with respect to the: x-axis y-axis origin

x-axis -> for every point (x, y) there is a point (x, -y) y-axis -> for every point (x, y) there is a point (-x, y) origin -> for every point (x, y) there is a point (-x, -y) i.e. if you rotate the graph 180⁰ the graph is unchanged (see image)

How will each graph be reflected in relation to the other? For y = f(x), draw the graph y = f(-x). For y = g(x), draw the graph y = -g(x).

y = f(-x) is reflected across the y-axis Take the endpoints of each segment and multiply x by -1 (-4, 5) (4, 1) and (4, 1) (7, 4) (4, 5) (-4, 1) and (-4, 1) (-7, 4) y = -g(x) is reflected across the x-axis Take the segments and multiply y by -1. (-4, -4) (4, -6) (-4, 4) (4, 6) * Watch out for graphs that look like they should be reflected one way, but that's not what the question is asking for.

How would the following graphs be transformed? y = f(x) -> y = f(2x) y = g(x) -> y = 1/2 g(x)

y = f(2x) - divide the x values of the coordinates by 2 y = 1/2 g(x) - divide the y values of the coordinates by 2 * Note the switch: x values receive opposite operator. Y values receive the same.

What is the end behavior of the following graphs? (i.e. falls to the left rises to the right, ...) f(x) = 6x⁵ - 2x⁴ + 4x³ + 2x g(x) = 6x⁴ + 6x³ - 3x² - 4x h(x) = x²(3x - 5)² i(x) = -2x⁵ - 3x³ + 9x² - 9x

Determined by the first term... ------------- - Even degree -> rises or falls on both sides - Odd degree -> rises on one side, falls on other - Positive coefficient -> falls to left and rises to right; or rises on left and right - negative coefficient -> rises to the left and falls to the right; or falls to left and right f(x) = 6x⁵ - 2x⁴ + 4x³ + 2x 5 -> odd, 6 -> positive ... - falls to the left, rises to the right g(x) = 6x⁴ + 6x³ - 3x² - 4x 4 -> even, 6 -> positive - rises on the left and right h(x) = x²(3x - 5)² = 9x⁴ ... 4 -> even, 9 -> positive - rises on the left and right i(x) = -2x⁵ - 3x³ + 9x² - 9x 5 -> odd, -2 -> negative - rises to the left, falls to the right

What is the button on the calculator that calcuates local minimums and maximums?

Extrema

Find the domain of the function and write in interval notation: g(x) = √(x - 9)

Find the set of all values that make √(x - 9) a real number (i.e. when x - 9 is greater than or equal to zero). x - 9 >= 0 x >= 9 [ 9, ∞) * bracket -> includes point parenthesis -> point not included (can't include infinity)

Rewrite the set N by listing its elements. Make sure to use the appropriate set notation. N = {y | y is an integer and -4 <= y < 0}

N = {-4, -3, -2, -1}

Solve x² = -24, where x is a real number. Simplify as much as possible.

No solution

What does it mean that a graph is one-to-one?

No two points on a graph have the same y-coordinate.

How would one go about solving the following system of equations: 5x + 7y = 17 -8x + 3y = -13

One must elimate one of the variables. You could eliminate the x by multiplying both sides of the top equation by 8 and the bottom by 5, resulting in: 40x + 56y = 136 -40x + 15y = -65 ------------------ 71y = 71 y = 1 5x + 7(1) = 17 5x = 10 x = 2

Use the Factor Theorem to determine whether x + 3 is a factor of P(x) = x⁴ + 2x³ − 2x - 9.

P(x) = x⁴ + 2x³ − 2x - 9 P(-3) = 81 - 54 + 6 - 9 P(-3) = 24 != 0 Not a factor

The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 4335 kg and decreases continuously at a relative rate of 12% per day. Find the mass of the sample after three days. Do not round any intermediate computations, and round your answer to the nearest tenth.

Peʳᵗ 4335 e ⁻⁰·¹² * ³ = 3024.4 kg

Graph the function: f(x) = { 2 if x != 0 { -1 if x = 0

Put points, draw line (not a segment or ray)

Value of R(d) of dollars in euros: R(d) = 9/10(d) Cost P(n) in dollars to purchase and ship n purses: P(n) = 99n + 23 Write formula for cost Q(n) in euros to purchase and ship purses.

Q(n) = 9/10 (99n + 23)

A cookie company uses one cup of sugar for every 30 cookies it makes. Let S represent the total number of cups of sugar used, and let N represent the number of cookies made. Write an equation relating S to N.

S = 30/N

Braking distance D(v) at certain velocity v: D(v) = v²/34 Velocity after t seconds: B(t) = 5t Formula for breaking distance S(t) after t seconds?

S(t) = (5t)²/34 * Don't forget the parentheses

Translate y = x² to: y = x² - 2 y = (x - 3)² on a graph.

See the attached photo for y = x² - 2. It is shifted down two units. If it were x² + 2 it would shift up. When the format is (x - 3)² the graph is shifted to the right three units. If it were (x + 3)² it would shift to the left. i.e. The opposite of what you would intuitively guess based on the vertical shifting rule stated above

Translate y = (1/2)ˣ into: y = (1/2)ˣ⁺⁴ + 1

Shift the line 4 units left and 1 unit up.

When looking for the absolute maximum and minimum in a function, how does a vertical asymptote effect the search?

The direction of the function at the asympote means there is no absolute maximum or minimum.

In a parabola, what happens to the lines as the coefficient gets closer to 0?

The graph lines become wider

Are these graphs continuous: 2x - 1 if x < 1 -x + 2 if x >= 1

Yes

Given the graph of function g (see screenshot), find: a) All local minimum values of g. b) All values at which g has a local minimum.

a) g = 0 b) -3, 3

Graph the function: f(x) = √x - 5 Find the leftmost point and three additional points.

f(x) = √x - 5 Leftmost point uses square root of 0. √0 - 5 -> (0, -5) (1, -4) (4, -3) (9, -2)

Find: g(5x) = x² -3

g(5x) = x² -3 = (5x)² - 3 = 25x² - 3

Solve for y. Round to the nearest hundredth. e⁶ʸ = 4

ln(e) = 1 e⁶ʸ = 4 lne⁶ʸ = ln4 6y(lne) = ln4 6y = ln4 y = ln4/6 y = 0.23

Solve for x. ln(x - 2) - ln 4 = -3

ln(x - 2) - ln 4 = -3 ln (x - 2)/4 = -3 (x - 2)/4 = e⁻³ x = 4e⁻³ + 2 x = 2.20

Solve for x: ln(x - 5) - ln14 = ln18

ln(x - 5) - ln14 = ln18 ln (x-5)/14 = ln 18 (x - 5)/14 = 18 x = 252 + 5 x = 257

Use the change of base formula to compute: log ₁⸝₇ 4 Round to the thousandth decimal place.

log 4 ------ log 1/7 -0.712

Solve for x: log(x + 4) = log(5x + 5)

log(x + 4) = log(5x + 5) x + 4 = 5x + 5 -1 = 4x x = -1/4

Expand: log(yz⁸)

logy + 8logz

Use the properties of logarithms to evaluate the following expression: log₁₄7 - log₁₄2 = __

log₁₄7 - log₁₄36 = __ logₐ(M/N) = logₐM - logₐNterm-75 logₐ(M*N) = logₐM + logₐN = log₁₄14 logₐb = c -> aᶜ = b so... logₐaᶜ = c = log₁₄14¹ = 1

Solve for x: log₄(-3x + 8) = 1

logₐb = c -> aᶜ = b 4¹ = -3x + 8 3x = 4 x = 4/3

What is a parabola? What is its vertex?

vertex = point at which parabola turns around

Two trains leave towns 542 kilometers apart at the same time and travel toward each other. One train travels 21 kmh faster than the other. If they meet in 2 hours, what is the rate of each train?

x = rate of faster train y = rate of slower train x - y = 21 y = x - 21 distance = rate x time 2(x) + 2(x - 21) = 542 4x = 584 x = 146 y = 146 - 21 y = 125

Write in terms of i. Simplify your answers as much as possible. √-45 -√-48 √-16

√-45 3i√5 -√-48 -4i√3 √-16 4i

Simplify: √5(√3 - 8)

√5(√3 - 8) √15 - 8√5 * Check for actual square roots

Simplify: √7c * √7c √3c * √2c

√7c * √7c 7c √3c * √2c √6c² c√6

What is the vertex of: -2(x + 1)² + 3

(-1, 3) Where x -> 0 -2( (-1) + 1 )² And y is what's left -> (0) + 3

Determine the interval(s) on which the function is (strictly) increasing.Write your answer as an interval or list of intervals.When writing a list of intervals, make sure to separate each interval with a comma and to use as few intervals as possible.Click on "None" if applicable.

(-6, -4), (1, 2)

Find the domain of the function: f(x) = (x + 9) / √(x - 10)

(10, ∞)

Factor: 9v² + 24v + 16

(3v + 4)(3v + 4)

Describe the following graph: 3x g(x) = ------- x² + 2x - 8

(x + 4)(x - 2) -> asymptotes at -4, 2 (-∞, -4) (-4, 2) (2, ∞) Higher degree on bottom (x²) so horizontal asymptote at 0 X values in ranges: g(-5) -> negative -> starts with negative asymptote g(-3) -> positive -> positive asymptote g(1) -> neg g(3) -> pos

A 5 in thick slice is cut off the top of a cube, resulting in a rectangular box that has volume 45 in³. Use the ALEKS graphing calculator to find the side length of the original cube. Round your answer to two decimal places.

(x)(x)(x - 5) = 45 x³ - 5x² -45 Find the y-intercept. x = 6.18 in.

Multiply and simplify: (√x - 2√3)²

(√x - 2√3)² (√x - 2√3)(√x - 2√3) x - 2√3x - 2√3x + 12 x - 4√3x + 12

What is the quadratic formula?

** It's 2a on the bottom. NOT 2b!!!!!

Two factory plants are making TV panels. Yesterday, Plant A produced 6000 panels. Two percent of the panels from Plant A and 6% of the panels from Plant B were defective. How many panels did Plant B produce, if the overall percentage of defective panels from the two plants was 5%?

.02(6000) + .06(x) = .05(6000 + x) x = 18,000

Simplify the following: 1) 125⁻²ᐟ³ 2) (1 / 8)⁻²ᐟ³

1) 125⁻²ᐟ³ 1 / 125²ᐟ³ 1 / (125¹ᐟ³)² 1 / (³√125)² 1 / 5² 1 / 25 2) (1 / 8)⁻²ᐟ³ 8 ²ᐟ³ (8¹ᐟ³)² (³√8)² 2² 4

Fill in the missing ( __ ) values: 1) log₈3 - log₈5 = log₈__ 2) log₉__ + log₉5 = log₉35 3) -4log₈3 = log₈__

1) logₐ(M/N) = logₐM - logₐN log₈(3/5) 2) logₐ(M*N) = logₐM + logₐN log₉7 3) logₐMᵖ = p logₐM log₈(1/81)

Evaluate each expression: 1) log ₅ 1/25 = x 2) log ₂ 32 = x

1) logₐc = b aᵇ = c 5ˣ = 1/25 5ˣ = 5⁻² x = -2 2) log ₂ 32 = x 2ˣ = 32 x = 5

Are the following sets of ordered pairs functions? 1) { (z, w), (j, c), (j, m), (k, r) } 2) { (b, c), (a, d), (c, c), (d, a) }

1) no, two "j"s as domains (i.e. "x") -> (j, c) and (j, m) 2) yes, all domains unique (b, a, c, d)

How would you translate the following graphs? 1) y = |x| -5 2) y = |x - 2|

1) y = |x| -5 - down 5 units 2) y = |x - 2| - right 2 units

Least common multiple of: 9m² and 6m 7n and 8c³

18m² 56nc³

Describe the following graph: x² + 1 f(x) = ------ 2x² - 8

2(x² -4) -> asymptotes == -2, 2 (-∞, -2) (-2, 2) (2, ∞) Same degree on top and bottom (x²) so horizontal asymptote at 1/2 Choose x values inside ranges: f(-3) = 1 -> positive asymptote f(0) = -1/8 -> negative f(3) = 1 -> positive

On a rectangular piece of cardboard with perimeter 17 inches, three parallel and equally spaced creases are made. (See Figure 1.) The cardboard is then folded along the creases to make a rectangular box with open ends. (See Figure 2.) Letting x represent the distance (in inches) between the creases, use the ALEKS graphing calculator to find the value of x that maximizes the volume enclosed by this box. Then give the maximum volume. Round your responses to two decimal places.

2h + 2(4x) = 17 h = (17-8x) / 2 volume = ((17-8x) / 2)(x)(x) Graph and find max x = 1.42 in y = 5.69 in ³

Solve for x with the quadratic formula: 2x² - 4x = 3

2x² - 4x - 3 = 0 -(-4) +- √(-4² - 4(-6)) ---------------------- 2(2) 4 +- √40 ---------- 4 x = 2.58, -0.58

Rewrite log₂16 = 4 as an exponential equation.

2⁴ = 16 logₐc = b -> aᵇ = c acb -> abc

Factor: 3x² + 27x -45

3(x² + 9x - 15) Cannot be factored

Write (3y³)² without exponents.

3*y*y*y*3*y*y*y

Find the least common multiple of: 15y and 6m²

30ym²

A loan of $34,000 is made at 7% interest, compounded annually. After how many years will the amount due reach $77,000 or more? Write the smallest possible whole number answer.

34,000 (1.07)ˣ >= 77,000 1.07ˣ >= 2.26 Try out x values until you get the smallest value that is greater than or equal to 2.26. x = 13

Simplify: 7 - 1/5v -------- 3 + 1/5v

35v - 1 ------- 15v + 1

Simplify: 8(3v + 5)(v - 7) ---------------- 48(v - 7)(2v + 5)

3v + 5 ------- 12v + 30

Graph all the vertical and horizontal asymptotes of: 3x² + 12x + 12 f(x) = --------------- -x² + 1

3x² + 12x + 12 f(x) = --------------- -x² + 1 Is it in its simplest form? 3(x² + 4x + 4) f(x) = --------------- -(x - 1)(x + 1) 3(x + 2)² f(x) = --------------- -(x - 1)(x + 1) Yes. Vertical asymptotes -> 0 in denominator x = 1, -1 Horizontal asymptotes -> degrees of num/den if num < den -> y = 0 if num > den -> none if num == den -> lead coefficient of num/ den ** 3x² / -x² -> 2 == 2 3 / -1 y = -3

Multiply: 3√6 (√2 - √7)

3√6 (√2 - √7) 3√12 - 3√42 (√12 -> √4√3 -> 2√3) 6√3 - 3√42

Write the quadratic equation whose roots are 2 and −6 , and whose leading coefficient is 4. (Use the letter x to represent the variable.)

4(x - 2)(x + 6) = 0 4x² + 16x - 48 = 0

A forest covers 4100 km². It's area will decrease by 8.75% every year. How large will it be after 15 years?

4100(1 - .0875)¹⁵

Simplify: 4u² - 20u ---------- 5u² - 25u

4u(u - 5) --------- 5u(u - 5) 4/5

Factor: 4y² + 36y + 81 16u² - 56u + 49

4y² + 36y + 81 (2y + 9)² 16u² - 56u + 49 (4u - 7)²

Multiply: 6√3 (√5 + √3)

6√3 (√5 + √3) 6√(3*5) + 6√(3*3) 6√15 + 6√9 6√15 + 18

For a function with 6 local extrema, what are the possible degrees of the function?

A function with a degree of n has at most n-1 extrema, so the possible degrees with 6 local extrema are 7, 9, 11, ...

Graph five points of the exponential function of f(x) = 4ˣ and include the asymptote.

(-2, 1/16) (-1, 1/4) (0, 1) (1, 4) (2, 16) Plot the asymptote as a horizontal line along the x-axis. Then graph-a-function.

Plot 2 points and the asymptote of: g(x) = (1/3)ˣ - 2 Give the domain and range in interval notation.

(0, -1) (-1, 1) (-2, 7) (1, -1.67) (2, -1.89) * approaching -2 domain = (-∞, ∞) range = (-2, ∞)

Find the x- and y-intercepts: -5x + 2y = 6

x-intercept -> y = 0 -5x + 2(0) = 6 -5x = 6 x = -6/5 y-intercept -> x = 0 -5(0) + 2y = 6 2y = 6 y = 3

Find all the values of x that are not in the domain of h. h(x) = x² + 2x - 24 ------------ x² + 8x + 12 h(x) = x² - 4x -45 ------------ x² - 81

x² + 8x + 12 (x + 6)(x + 2) = 0 x = -6, -2 x² - 81 x = 9, -9

Give an explanation of how the following logarithms work and solve them. log₂8 log₄4 log₄1

log₂8 2 to which power equals 8? log₂8 = 3 log₄4 4 to which power equals 4? log₄4 = 1 log₄1 4 to which power equals 1? log₄1 = 0

Solve for x: log₄(x - 8) - log₄2 = log₄x

log₄(x - 8) - log₄2 = log₄x (x - 8)/2 = x x - 8 = 2x x = -8 Check: log₄(-8 - 8) log₄(-16) = undefined -> No solution

Expand the following equation. Only one variable per log and no radicals or exponents. ³ √ x⁵z log √ ------ √ y² * see image

x⁵z ¹ᐟ³ log ---- y² x⁵z (1/3)log ---- y² (1/3) (logx⁵z - logy²) (1/3) (5logx + logz - 2logy) (5/3)logx + (1/3)logz - (2/3)logy

Write the expression as a single logarithm. (1/4)log꜀w + 4log꜀x - log꜀y

(1/4)log꜀w + 4log꜀x - log꜀y log꜀w¹ᐟ⁴ + log꜀x⁴ - log꜀y log꜀ (w¹ᐟ⁴x⁴ / y)

Suppose H(x) = (3x + 4)⁶ Find two functions f and g such that (f * g)(x) = H(x)

(f * g)(x) = f(g(x)) g(x) = 3x + 4 f(x) = x⁶

Define the new functions of: r(x) = x² s(x) = 5x³ Given the expressions: (s + r)(x) (s * r)(x) (s - r)(-1)

(s + r)(x) -> s(x) + r(x) (s * r)(x) -> s(x) * r(x) (s - r)(-1) -> s(-1) - r(-1)

Simplify: w² - 2w - 15 ------------- 5w² - 45

(w - 5)(w + 3) -------------- 5(w² - 9) (w - 5)(w + 3) -------------- 5(w - 3)(w + 3) w - 5 ------- 5(w - 3)

Rewrite the expression by factoring out (w - 8): 7w²(w - 8) - 2(w - 8)

(w - 8)(7w² - 2)

Solve for w: (w² -9)(5w - 40) = 0

(w² -9)(5w - 40) = 0 w = 3, -3, 8

Write a quadratic function whose zeros are -2 and -11.

(x + 2)(x + 11) * Don't expand

Write a quadratic function whose only zero is -2.

(x + 2)(x + 2)

Use synthetic division to find the quotient and remainder when (6x⁴ + 8x³ - 4x² + 17x + 18) is divided by (x + 2). Write the remainder as: Remainder ------------ x + 2

* Synthetic division is a shortcut used for dividing a polynomial of degree 1 or higher by a polynomial of the form (x - r). ** Write 0 as the coefficient for any missing powers. *** If dividing by (x - 7), the divisor becomes positive (i.e. 7).

Solve for x: 3 ⁻⁹ˣ = 11 ˣ⁺⁸

-9xlog3 = (x + 8)log11 -9xlog3 = xlog11 + 8log11 x(-9log3 - log11) = 8log11 x = 8log11 / (-9log3 - log11)

Divide. Simplify as much as possible. 2x + 18 4x + 36 ------- ÷ --------- x - 6 3x - 18

2x + 18 4x + 36 ------- ÷ --------- x - 6 3x - 18 2(x + 9) 3(x - 6) ------- x -------- x - 6 4(x + 9) 2 * 3 3 ----- = --- 4 2

What are the quotient and remainder of: (2x² + 8x + 10) ÷ (x + 2)

2x + 4 _________________ x + 2 √ 2x² + 8x + 10 2x² + 4x --------- 4x + 10 4x + 8 ------- 2 Quotient = 2x + 4 Remainder = 2 * Remainders can be negative

Use the quadratic formula to solve for x. 2x² + 8x + 1 = 0

2x² + 8x + 1 = 0 -8 +- √(8² - 4(2)) ----------------- 4 (-8 +- √56) ÷ 8 (-8 +- 2√14) ÷ 8 (-4 +- √14) ÷ 2

Graph the rational function: 2x² - 6x g(x) = --------- x² - 4x + 3

2x² - 6x g(x) = --------- x² - 4x + 3 2x(x - 3) g(x) = --------- (x - 1)(x - 3) 2x = --------- x - 1 vertical asymptote -> 1 horizontal asymptote -> 2 hole -> 3 (i.e x not defined at 3) points -> (0, 0) (2, 4)

Solve for x. Round to the nearest hundredth. 5³ˣ = 4

5³ˣ = 4 log5³ˣ = log4 3xlog5 = log4 3x = log5 / log4 x = log5 / 3log4 x = 0.29

A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function C(x) = 0.2x² - 56x + 12,078. How many machines must be made to minimize the unit cost? Do not round your answer.

C(x) = 0.2x² - 56x + 12,078 x = - (-56 / 2(0.2)) = 140 * Watch out for what the question asks for. I.e. Don't just solve for the vertex's C(x) in this one, when all it wants is the axis of symmetry.

What is the difference between finding the asymptotes of compositions (f ⁰ g) and adding them (f + g)? f(x) = 4 / (x - 3) g(x) = 1 / x

Compositions only find the asymptotes from the equation created: e.g. f(g(x)) f(x) = 4 / (x - 3) g(x) = 1 / x f(g(x)) = 4 / (1/x - 3) (-∞, 0) u (0, 1/3) u (1/3, ∞) No 3 from original f(x) Adding, subtracting, etc. the two equations involves using the original equations and the result equation to find asymptotes.

What are the degrees of the following equations: 6y²z⁵ 2t³ + 1 - t⁵

For a monomial, the degree is the sum of the exponents: 2 + 5 = 7 For a polynomial, it's the highest degree of its terms: 5 * If the degree of a polynomial is 2, then it's quadratic (ax² + bx + c).

Factor the following expression: 12uv⁴x² - 28u⁷v⁹

GCF = 4uv⁴ 4uv⁴(3x² - 7u⁶V⁵)

When looking for the absolute maximum and minimum in a function, what happens if there is a hole at either location.

There is no absolute maximum or minimum.

Graph the vertical and horizontal asymptotes of: f(x) = 2 / (x + 2)

Vertical asymptote is at zero(s) of denominator. Horizontal asymptote (if any) compares degree of numerator (n) with degree of denominator (m). n < m -> y = 0 n = m -> y = lead coefficient n / lead coefficient of m n > m -> none f(x) = 2 / (x + 2) x = -2 2 / (x + 2) -> 0 / 1 y = 0

Considering the attached graph, what are the: - vertical and horizontal asymptotes - x- and y-intercepts - domain and range

Vertical asymptote: x = 2 Horizontal asymptote: y = -1 x-intercept: -6 y-intercept: 3 Domain: (-∞, 2)u(2, ∞) Range: (-∞, -1)u(-1, ∞)

When does an equation have a slant asymptote?

When the degree of the numerator is only one degree higher than the denominator, which also means there is no horizontal asymptote.

An initial population of 10 fish is introduced into a lake. This fish population grows according to a continuous exponential growth model. There are 28 fish in the lake after 11 years. (a) Create a formula for this growth. y = _e-ᵗ (b)How many fish are there 21 years after the initial population is introduced? Do not round any intermediate computations, and round your answer to the nearest whole number.

a) y = _e-ᵗ 28 = 10e¹¹ʳ 2.8 = e¹¹ʳ ln 2.8 = 11r r = ln 2.8 / 11 y = 10e⁽ˡⁿ²·⁸ ᐟ ¹¹⁾ * ᵗ b) 10e ⁽ˡⁿ²·⁸ ᐟ ¹¹⁾ * ²¹ = 71 fish

Complete the square for: f(x) = 2x² + 8x - 5 g(x) = 2x² + 16x + 33

f(x) = 2x² + 8x - 5 = 2(x² + 4x) - 5 -> (4 / 2)² = +4, -4 = 2(x² + 4x +4) - 5 + (2)(-4) = 2(x + 2)² - 13 g(x) = 2x² + 16x + 33 = 2(x² + 8x) + 33 -> (8 / 2)² = 16, -16 = 2(x² + 8x + 16) + 33 + (2)(-16) = 2(x + 4)² + 1

Find the domain of: f(x) = ln√(2x + 7) Write your answer as an interval or union of intervals.

f(x) = ln√(2x + 7) 2x + 7 > 0 2x > -7 x > -7/2 (-7/2, ∞)

Graph the logarithmic function: f(x) = log ₁⸝₃ x

f(x) = log ₁⸝₃ x 1/3 ʸ = x Choose powers of 3 to make it easier. x = 9 (1/3) ʸ = 9 (1/3) ⁻² = 9 (9, -2) (1/3) ⁻¹ -> (3, -1) (1/3) ⁰ -> (1, 0) (1/3) ¹ -> (1/3, 1) (1/3) ² -> (1/9, 2)

Find the domain of: f(x) = log(1 - 4x) Write your answer as an interval or union of intervals.

f(x) = log(1 - 4x) 1 - 4x > 0 -4x > -1 x < 1/4 (-∞, 1/4) * switch sign when dividing by negative number (-4)

Find the domain of: f(x) = log₃(x² - 1) Write your answer as an interval or union of intervals.

f(x) = log₃(x² - 1) 3ʸ = x² - 1 x² - 1 > 0 x < -1 or x > 1 (-∞, -1)u(1, ∞)

Find the y- and x-intercepts of: f(x) = x³ + x² - 16x -16

f(x) = x³ + x² - 16x -16 x(x² - 16) + (x² = 16) (x + 1)(x - 4)(x + 4) x-intercepts = -1, -4, 4 y-intercept = -16

Find the domain of the functions: f(x) = √(-3 + x) / (-6 + x) g(x) = -8 / √(x + 10)

f(x) = √(-3 + x) / (-6 + x) √(-3 + x) -> [3, ∞) (-6 + x) -> [3, 6)U(6, ∞) g(x) = -8 / √(x + 10) [-10, ∞)

Choose the graph that best fits the following functions: g(x) = -3(x + 1)²(x + 3)² f(x) = x³ - x² -6x

g(x) = -3(x + 1)²(x + 3)² Leading element == -3x⁴ -> falls to the left and right (D or E) Zeros == -1, -3 -> (still D or E) -1 has multiplicity of 2 -> touches but doesn't cross x-axis Same for -3 -> graph D f(x) = x³ - x² -6x x³ -> falls to the left and rises to right -> (B or F) Zero -> (0, 0) -> graph B ... Taking it further x(x - 3)(x + 2) Zeros -> 0, 3, -2 x = 0 -> multiplicity of 1 -> odd -> crosses x-axis Same for 3, -2 -> all cross x-axis

Graph the logarithmic function: g(x) = log₃(x + 1) + 2 To do this, plot two points on the graph of the function, and also draw the asymptote. Then, click on the graph-a-function button. Additionally, give the domain and range of the function using interval notation.

g(x) = log₃(x + 1) + 2 3ʸ = x (3, 1) (9, 2) (1, 0) (1/3, -1) Subtract 1 from x (move function left 1) Add 2 to y (move function up 2) (2, -1) (8, 0) (0, -2) (-2/3, 1) Asymptote at -1 Domain: (-1, ∞) Range: (-∞, ∞)

Write equations for the horizontal and vertical lines passing through the point (1, 8).

horizontal -> y = 8 vertical -> x = 1 * Pay attention to horizontal and vertical labels

Solve for x: log₄(x - 2) = 1 - log₄(x - 5)

log₄(x - 2) = 1 - log₄(x - 5) log₄(x - 2)(x - 5) = 1 log₄(x² - 7x + 10) = 1 x² - 7x + 10 = 4¹ x² - 7x + 6 = 0 (x - 6)(x - 1) = 0 x = 6, 1 - Check x = 6 log₄(6 - 2) = 1 - log₄(6 - 5) log₄4 = 1 - log₄1 1 = 1 - 0 true x = 1 log₄(-1) = 1 - log₄(-4) false - So... x = 6

Solve for x: log₄(x - 5) = log₄(x - 2) + 1

log₄(x - 5) = log₄(x - 2) + 1 log₄( (x-5)/(x-2) ) = 1 (x-5)/(x-2) = 4¹ x - 5 = 4x - 8 3 = 3x x = 1 log₄(-4) = log₄(-1) + 1 -> no solution

Given the functions: u(x) = 2x w(x) = -2x² + 1 Find the following: (u ∘ w)(-1) (w ∘ u)(-1)

u(x) = 2x w(x) = -2x² + 1 * The composition of u with w. (u ∘ w)(-1) = u(w(-1)) = u(-1) = 2(-1) = -2 (w ∘ u)(-1) = w(u(-1)) = w(-2) = -7

Find the equation of the quadratic function g whose graph has a vertex of (3, 4) and a point (5, -4).

vertex = (3, 4) g(x) = a(x - 3)² + 4 (5, -4) -4 = a(5 - 3)² + 4 -4 = 4a + 4 a = -2 g(x) = -2(x - 3)² + 4

Graph all asymptotes of: x² - x + 8 f(x) = ---------- x - 3

vertical -> x = 3 horizontal -> none slant -> yes, x² one degree higher than x x + 2 + 14 / (x - 3) x - 3 √ x² - x + 8 y = x + 2

Find all real zeros of the function. f(x) = -4x(x² + 49)(x² -1)

x = 0, 1, -1

Find all real zeros of the function. g(x) = 3(x - 3)(x + 3)²(x² - 16)

x = 3, -3, 4, -4

Solve x² = 27, where x is a real number. Simplify as much as possible.

x = √27 or -√27 √3√9 -> 3√3 x = 3√3 or -3√3

Solve the following where x is a real number. x³ = 5 x³ = -10 x³ = -9

x³ = 5 x = ³√5 x³ = -10 x = ³√-10 x = ³√(-1)³ * 10 x = - ³√10 x³ = -9 x = - ³√9 * resist the tempation to think of this as x²`

Fill in the blank: x * x⁷ = x- y⁵ * y² = y-

x⁸ y⁷

Are the absolute maximum and minimum values of a function x or y coordinates?

y (see abs max of image)

Graph the equation: y = 5 |x + 5|

y = 5 |x + 5| vertex -> |x + 5| = 0 = -5 (-5, 0) (-6, 5) (-4, 5)

For the polynomial below, 3 is a zero: f(x) = x³ + 3x² - 12x - 18 Express f(x) as a product of linear factors.

zero -> (x - 3) Use synthetic division: __________________ 3 √ 1 + 3 - 12 - 18 3 18 18 ---------------- 1 6 6 0 so -> (x-3) (x² + 6x + 6) -6 +- √(36 - (4)(6)) --------------------- = -3 + √3 2(1) (x-3) (x - (-3 + √3)) (x - (-3 - √3))

Given: g = {(-6, 7), (-3, -2), (4,-6), (7, 3)} h(x) = 2x + 3 Find the following: g⁻¹(7) h⁻¹(x) (h * h⁻¹)(-7)

⁻¹ == the inverse g⁻¹(7) Reverse the x and y {(7, -6), (-2, -3), ...} h⁻¹(x) y = 2x + 3 -> x = 2y + 3 y = (x - 3) / 2 (h * h⁻¹)(-7) h(h⁻¹(-7)) h⁻¹(-7) = (-7 - 3) / 2 = -5 h(-5) = 2(-5) + 3 = -7

The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of 7.9% per hour. How many hours does it take for the size of the sample to double? Note: This is a continuous exponential growth model. Do not round any intermediate computations, and round your answer to the nearest hundredth.

A = Peʳᵗ 2P = Pe⁰·⁰⁷⁹* ᵗ 2 = e⁰·⁰⁷⁹* ᵗ ln 2 = ln (e⁰·⁰⁷⁹* ᵗ) ln 2 = 0.079t t = ln 2/ 0.079 t = 8.77 hours

Suppose that $2700 is borrowed for three years at an interest rate of 8% per year, compounded continuously. Find the amount owed, assuming no payments are made until the end. Do not round any intermediate computations, and round your answer to the nearest cent.

A = Peʳᵗ A = 2700e⁽⁰·⁰⁸⁾⁽³⁾

Use the properties of logarithms to evaluate the following expression: ln e⁸ + ln e⁴ = __

ln c = b -> eᵇ = c so... ln eᵇ = b ln e⁸ + ln e⁴ = 8 + 4 = 12

Latoya places a bottle of water inside a cooler. As the water cools, its temperature Ct in degrees Celsius is given by the following function, where t is the number of minutes since the bottle was placed in the cooler. C(t) = 3 + 21e⁻⁰·⁰⁵ᵗ Latoya wants to drink the water when it reaches a temperature of 20 degrees Celsius. How many minutes should she leave it in the cooler? Round your answer to the nearest tenth, and do not round any intermediate computations.

C(t) = 3 + 21e⁻⁰·⁰⁵ᵗ 20 = 3 + 21e⁻⁰·⁰⁵ᵗ 17 = 21e⁻⁰·⁰⁵ᵗ 17/21 = e⁻⁰·⁰⁵ᵗ ln 17/21 = -0.05t ln e ln 17/21 / -0.05 = t (1) t = 4.2 min

A delivery truck is transporting boxes of two sizes: large and small. The combined weight of a large box and a small box is 90 pounds. The truck is transporting 50 large boxes and 70 small boxes. If the truck is carrying a total of 5100 pounds in boxes, how much does each type of box weigh? How would one go about solving this with the ALEKS graphing calculator?

Create two equations that solve for y and find the intersection. y = 90 - x 50x + 70x = 5100 y = (5100 - 50x) / 70 Plot these in "y=". Set the window size and auto calculate. Click intersection and then find intersection. x = 60 (large) y = 30 (small)

How to you compute simple interest?

interest = rate x time period x principal e.g. Interest on 20,000 loan, 5% per year, 5 years: interest = .05 x 5 x 20,000

Hong deposited $4000 into an account with 3% interest, compounded semiannually. Assuming that no withdrawals are made, how much will he have in the account after 8 years? Do not round any intermediate computations, and round your answer to the nearest cent.

Semiannual means every six months (i.e. twice per year). The annual rate is 3%, so half the rate gets compounded every six months. Rate per compound period = 1 + (0.03 / 2) Number of periods = 8 * 2 = 16 Amount = 4000 * (1 + (0.03/2) )¹⁶ = $5075.94

What is the solution to the following equations: x - 3y = -6 -x + 3y = -6

This system has no solution. 0 = -6

What are the leading term, degree, and leading coefficient of: 23v⁶ + v⁴ + 7v⁹ - 4v

leading term = 7v⁹ degree = 9 leading coefficient = 7

Write the domain and range of this graph using interval notation.

domain = all x coordinates -3 <= x < 3 domain = [ -3, 3) range = all y coordinates -5 <= y < 5 range = [ -5, 5) * bracket -> includes point parenthesis -> point not included

Write the domain and range of the following graph using interval notation.

domain = all x coordinates domain = [-2, 3) range = all y coordinates range = [-5, 4] * bracket -> includes point parenthesis -> point not included

How do you find the vertexes of these parabolas? y = -2x² + 4x + 10 y = (y - 3)² + 2

y = -2x² + 4x + 10 vertex = (1, 12) First solve for the axis of symmetry (the vertical line that divides the parabola in half): For: y = ax² + bx + c Axis of symmetry = - b/2a - (4/((2)-2) = - (4/-4) = 1 The axis of symmetry is the x coordinate of the vertex, so then just solve for y. -2(1)² + 4(1) + 10 (1, 12) ------------ y = (y - 3)² + 2 vertex = (3, 2) For: y = (x - h)² + k Vertex = (h, k) Note the sign switch of h.

The amount of money invested in a certain account increases according to the following function, where y₀ is the initial amount of the investment, and y is the amount present at time t (in years). y = y₀(e⁰·⁰³⁵ᵗ) After how many years will the initial investment be doubled? Do not round any intermediate computations, and round your answer to the nearest tenth.

y = y₀e⁰·⁰³⁵ᵗ 2y₀ = y₀e⁰·⁰³⁵ᵗ 2 = e⁰·⁰³⁵ᵗ ln 2 = ln (e⁰·⁰³⁵ᵗ) ln 2 = 0.035t t = ln 2 / 0.035 t = 19.8 years

Solve for x: (x - 5)² = 2x² - 7x -3

(x - 5)² = 2x² - 7x -3 = 2x² - 7x -3 - (x - 5)² ---------- - (x - 5)² -( (x - 5)(x - 5) ) - (x² - 10x + 25) ---------- = 2x² - 7x -3 - x² + 10x - 25 = x² + 3x - 28 = (x + 7)(x - 4) x = -7, 4

Solve the inequality for w: -22 - 5w < 3

-22 - 5w < 3 -22 + 22 - 5w < 3 + 22 -5w < 25 w > -5 * When dividing both sides by a positive number the inequality sign stays the same. When dividing both sides by a negative number, the sign is reversed.

For f(x) = (1/6)ˣ find: x = -3, -2, -1, 0, 1

-3 -> 216 -2 -> 36 -1 -> 6 0 -> 1 1 -> 1/6

Solve for x: -6log₄(5x) = -12

-6log₄(5x) = -12 log₄(5x) = -12/-6 log₄(5x) = 2 logₐb = c -> aᶜ = b 4² = 5x 16 = 5x x = 16/5

Find the domain of: f(x) = √(-8x + 48) Write the answer in interval notation.

-8x + 48 >= 0 -8x >= -48 x = 6 x = 7 -> not a real number √(-56 + 48) x = 5 -> ok, so domain must decrease from 6 (-∞, 6]

Convert the following to positive exponents: v³ * v⁻⁸ * v

1 / v⁴ 3 - 8 + 1 = -4

How would you translate the following graphs? 1) y = f(x) -2 2) y = g(x - 5)

1) y = f(x) -2 - down 2 units 2) y = g(x - 5) - right 5 units

A laptop computer is purchased for $1700 . Each year, its value is 75% of its value the year before. After how many years will the laptop computer be worth $600 or less?

1700 * 0.75ˣ <= 600 0.75ˣ <= 600/1700 0.75ˣ <= 0.3529 Try out values until you get the smallest number that results in less than or equal to 0.3529. x = 4

Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 2000 bacteria selected from this population reached the size of 2140 bacteria in three hours. Find the hourly growth rate parameter. Note: This is a continuous exponential growth model. Write your answer as a percentage. Do not round any intermediate computations, and round your percentage to the nearest hundredth.

2000 e³ʳ = 2140 e³ʳ = 1.07 3r = ln 1.07 r = ln 1.07 / 3 r = 0.0226 * Write as a PERCENTAGE 2.26 %

Suppose that $2300 is invested at an interest rate of 4.25% per year, compounded continuously. After how many years will the initial investment be doubled? Do not round any intermediate computations, and round your answer to the nearest hundredth.

2300 e ⁰·⁰⁴²⁵ᵗ = 4600 e ⁰·⁰⁴²⁵ᵗ = 2 0.0425t = ln 2 t = ln 2 / 0.0425 t = 16.31 years

Use the properties of logarithms to evaluate the following expression: 2log₁₂4 + log₁₂9 = __

2log₁₂4 + log₁₂9 = __ logₐMᵖ = p logₐM log₁₂16 + log₁₂9 logₐ(M/N) = logₐM - logₐN logₐ(M*N) = logₐM + logₐN log₁₂(16 * 9) log₁₂(144) logₐb = c -> aᶜ = b so... logₐaᶜ = c log₁₂12¹² = 12

Use the properties of logarithms to evaluate the following expression: 2log₂3 - log₂36 = __

2log₂3 - log₂36 = __ logₐMᵖ = p logₐM 2log₂3 - log₂36 = log₂3² - log₂36 = log₂9 - log₂36 logₐ(M/N) = logₐM - logₐN logₐ(M*N) = logₐM + logₐN = log₂(9/36) = log₂(1/4) logₐb = c -> aᶜ = b so... logₐaᶜ = c = log₂2⁻² = -2

Write this expression as a single logarithm. 3log₄z + 4(log₄x - 5log₄y)

3log₄z + 4(log₄x - 5log₄y) log₄z³ + log₄x⁴ - log₄y²⁰ log₄z³ + log₄(x⁴ / y²⁰) log₄ (z³x⁴ / y²⁰)

A principal of $4,200 is invested at 7.75% interest compounded annually. How much will the investment be worth after 12 years?

4200(1.0775)¹² = $10,286

Solve for x. Round to the nearest hundredth. 5 - 2 ln x = 2

5 - 2 ln x = 2 -2 ln x = -3 ln x = 3/2 x = e³ᐟ² x = 4.48

Henry deposited $5000 into an account with a 7.2% annual interest rate, compounded monthly. Assuming that no withdrawals are made, how long will it take for the investment to grow to $7645 ? Do not round any intermediate computations, and round your answer to the nearest hundredth.

5000 (1 + 0.072/12)¹²ᵗ = 7645 1.006¹²ᵗ = 7645/5000 12t log 1.006 = log 1.529 12t = log 1.529 / log 1.006 t = log 1.529 / 12 log 1.006 t = 5.92 years * Don't multiply 5000 and 1.006 first i.e. 5030¹²ᵗ = 7645

Factor by grouping: 5y³ + 7y² + 25y + 35

5y(y² + 5) + 7(y² + 5) (5y + 7)(y² + 5)

Find the greatest common factor of these two expressions: 30x⁸w⁴ 18x³u⁷w⁶

6x³w⁴ Start with GCF of coefficients 30 and 18, which is 6. Then find the lowest powers of the common variables.

Expand: y⁷ log ---- x

7logy - logx

Write the expression as a single logarithm. 7logₘ(5z + 1) + (1/4)logₘ(z + 3)

7logₘ(5z + 1) + (1/4)logₘ(z + 3) logₘ(5z + 1)⁷ + logₘ(z + 3)¹ᐟ⁴ logₘ((5z + 1)⁷(z + 3)¹ᐟ⁴)

Solve for x: 81 = 27⁻ˣ⁺⁴

81 = 27⁻ˣ⁺⁴ 3⁴ = 3³⁽⁻ˣ⁺⁴⁾ 3⁴ = 3⁻³ˣ⁺¹² 4 = -3x + 12 3x = 8 x = 8/3

How do you simplify: (-3b³a²)⁴

81b¹²a⁸ -3 x -3 x -3 x -3 = 81 3 x 4 = 12 2 x 4 = 8

Simplify: sqrt{64x^4} (see photo)

8x²

Two trains leave the station at the same time heading west and east. The westbound train travels 95 mph. The eastbound train travels 85 mph. How long will it take them to travel 396 miles apart?

95x + 85x = 396 2.2 hours

Solve for y: 30y - 25 = 9y²

9y² - 30y + 25 = 0 9(25) = 225 -> -15 x -15 = 225 and -15 - 15 = -30 9y² - 15y - 15y +25 3y(3y - 5) - 5(3y - 5) (3y - 5)(3y - 5) = 0 (3y - 5) = 0 -> y = 1 2/3

Find all excluded values for: (v-4) / (v+2)

v = -2 All values where expression becomes undefined, i.e. can't divide by 0

Find the domain of: v(x) = √(-x) - 7 Use interval notation.

v(x) = √(-x) - 7 √(-x) >= 0 x = 0 (-∞, 0]

Find all excluded values of: x² + 3x - 18 ------------ x² - 49

excluded values = where denominator equals 0 x = -7, 7

Solve for g: f = (1/7)(g + h - k)

f = (1/7)(g + h - k) 7f = g + h - k g = 7f - h + k

If the graph g(x) = 5x² - 1 is translated vertically downward by 9 units, it becomes the graph of function h. Find h(x).

h(x) = g(x) - 9 = (5x² - 1) -9 = 5x² - 10

What is the solution to the following equations: x - 2y = -4 -x + 2y - 4 = 0

There are infinite solutions. -4 = -4 y = (x + 4)/2

Rewrite 5² = 25 as a logarithmic equation.

log₅25 = 2 aᵇ = c -> logₐc = b abc -> acb

Solve for x: log ₓ 1/36 = 2

logₐb = c aᶜ = b log ₓ 1/36 = 2 x² = 1/36 x = 1/6

Rewrite the following as an exponential equation: ln6 = y

logₐc = b -> aᵇ = c When the base is 'e', we don't write logₑ. We write 'ln' (aka natural log). 'e' is a special irrational number that equals 2.718281... ln c = b -> eᵇ = c eʸ = 6

Rewrite the following as a logarithmic equation: e⁵ = x

logₐc = b -> aᵇ = c When the base is 'e', we don't write logₑ. We write 'ln' (aka natural log). 'e' is a special irrational number that equals 2.718281... ln c = b -> eᵇ = c ln x = 5

Expand the following equation. Each log should only have one variable and no radicals or exponents. log√(x³yz⁵)

log√(x³yz⁵) log x³ᐟ² y¹ᐟ² z⁵ᐟ² 1/3logx + 1/2logy +5/2logz

What is a relation? Domain? Range? Function?

relation = set of ordered pairs consisting of a first and second component domain = set of first components (e.g. "x") range = set of second components (e.g. "y") function = special type of relation where no two ordered pairs have the same first component (i.e. domains must all be unique)