Updated CLEP Algebra Test

What are the steps to graphing polynomials?

1) Plug in ± values for x (start with 0 and go outward from there) and solve for y 2) Plot these points on the graph

Graph. y = x + 3 when x ≥ 1 { y = 2x - 1 when x < 1

1a) f(x) = x + 3 f(1) = 1 + 3 f(1) = 4 2a) Plot a closed circle at (1,4) with a diagonal arrow pointing in the positive direction (because of the ≥ sign) 1b) f(x) = 2x - 1 f(0) = 2(0) - 1 f(0) = -1 2b) Plot an open circle at (0,-1) with a diagonal arrow pointing in the negative direction (because of the < sign)

Graph. y = -3x - 8 when -4 ≤ x ≤ 0 { y = -3x + 9 when 0 < x ≤ 3

1a) ⋅f(x) = -3x - 8 f(-4) = -3(-4) - 8 f(-4) = 4 ⋅f(x) = -3x - 8 f(0) = -3(0) - 8 f(0) = -8 1b) Plot a point with a closed dot at (-4,4) and (0,-8), and connect the dots with a line 2a) ⋅f(x) = -3x + 9 f(0) = -3(0) + 9 f(0) = 9 ⋅f(x) = -3x + 9 f(3) = -3(3) + 9 f(3) = 0 2b) Plot a point with an open circle at (0,9) and with a closed circle at (3,0), and connect the dots with a line

Quadratic equations are order ___ polynomials.

2

Quadratic equations usually have ___ distinct root(s) / zero(es).

2

Simplify. 2 / (2i + 4)

2 / (2i + 4) [2 / (2i + 4)][(-2i + 4)/(-2i + 4)] (-4i + 8) / (-4i^2 + 16) (-4i + 8) / [(-4(-1) + 16] (-4i + 8) / (4 + 16) (-4i + 8) / 20 = (-i + 2) / 5

What is the formula to find the discriminant of a quadratic?

b^2 - 4ac

Things that would be represented well by exponential modeling are __________ things.

biological

Suppose the cost of building widgets is $20 per widget plus a fixed start up cost of $1,000. Write a function to represent the total cost.

c(x) = 20x + 1000

Solve. f(f(1)) when f(x)=x+2

f(1) = 1 + 2 f(1) = 3 f(3) = 3 + 2 ANSWER: f(3) = 5

Is f(x)=-x^3 symmetric? If so, what kind of symmetry does it have?

f(x) = -f(x) -x^3 = -(-x)^3 -2^3 = -(-2)^3 -8 ≠ 8 f(x) = f(-x) -x^3 = (--x)^3 -2^3 = (--2)^3 -8 ≠ 8 is not symmetric

Solve. f(x) = 3(x^3) - 1 when x=1

f(x) = 3(x^3) - 1 f(1) = 3(1^3) - 1 f(1) = 3 - 1 ANSWER: f(1) = 2

Write the inverse function of f(x) = 4x - 3

f(x) = 4x - 3 x = 4y - 3 4y = x + 3 ANSWER: y = (1/4)x + (3/4)

Is f(x)=x^2-4 symmetric? If so, what kind of symmetry does it have?

f(x) = f(-x) x^2 - 4 = (-x)^2 - 4 2^2 - 4 = (-2)^2 - 4 4 - 4 = 4 - 4 0 = 0 ANSWER: yes; has even symmetry

Is f(x)=x^3-7x symmetric? If so, what kind of symmetry does it have?

f(x) = f(-x) x^3 - 7x = (-x)^3 - 7(-x) (1)^3 - 7(1) = (-1)^3 - 7(-1) 1 - 7 = -1 - -7 -6 ≠ 6 NOT EVEN f(x) = -f(x) x^3 - 7x = -(x)^3 - 7 ⋅ -(x) 1^3 - 7(1) = -(1)^3 - 7 ⋅ -(1) 1 - 7 = -1 + 7 -6 = -6 yes; has odd symmetry

Is f(x)=x^4+1 symmetric? If so, what kind of symmetry does it have?

f(x) = f(-x) x^4 + 1 = (-x)^4 + 1 2^4 + 1 = (-2)^4 + 1 16 + 1 = 16 + 1 17 = 17 yes; has even symmetry

Write the original function of the inverse function, f^(-1)(x)=1/(x-1)

f^(-1)(x) = 1 / (x-1) x = 1 / (y-1) (y-1)x = 1 xy - x = 1 xy = 1 + x y = (1+x) / x

What is the format of an arithmetic series?

(nΣk=1)(ak+b)

Finite Geometric Series Format

(nΣk=1)(r^k) =

Write a function to represent the radius of a circle, given that A=π(r^2).

A(r) = π(r^2)

Which of the following products is a real number? a: (1 + i)((1 + i) b: (1 + i)(1 - i) c: (-1 - i)(-1 - i) d: (-1 + i)(-1 + i) e: (-1 + i)(-1 - i)

(1 + i)(1 - i) 1 - i + i -i^2 1 -(-1) 1 + 1 2 ANSWER: b

Simplify. (1 + i)^2(1 - i)

(1 + i)^2(1 - i) (1 + i)(1 + i)(1 - i) (1 + i + i + i^2)(1 - i) (1 + 2i + -1)(1 - i) 2i(1 - i) 2i - 2i^2 2i - 2(-1) = 2 + 2i

Compute. (1/3) + (7/5)

(1/3) + (7/5) (5+21)/15 = 26/15

Solve. (100Σk=1)(k) = ?

(100Σk=1)(k) 100[(1 + 100) / 2)] 100(101 / 2) = 5050

Solve. (10Σn=1)(1/2)^n

(10Σn=1)(1/2)^n [(1/2)(1 - (1/2)^10)] / [1 - (1/2)] [(1/2)(1 - (1/1024)] / (1/2) 1 - (1/1024) 1023 / 1024 ≈ 1

Simplify. (2^-3)(2^3)

(2^-3)(2^3) 2^0 = 1

Simplify. (2^4)(2^-1)^3

(2^4)(2^-1)^3 (2^4)(2^-3) 2^1 = 2

Simplify. (2i - 1) / (3i + 2)

(2i - 1) / (3i + 2) [(2i - 1) / (3i + 2)][(-3i + 2) / (-3i + 2)] (-6i^2 + 7i - 2) / (-9i^2 + 4) [-6(-1) + 7i - 2] / [-9(-1) + 4] (6 + 7i - 2) / (9 + 4) = (4 + 7i) / 13

When (3 + 4i) / (2 + i) is expressed in the form of a + bi, what is the value of "a"?

(3 + 4i) / (2 + i) [(3 + 4i) / (2 + i)][(2 - i) / (2 - i)] (6 + 5i - 4i^2) / (4 - i^2) [6 + 5i - 4(-1)] / [4 - (-1)] (6 + 5i + 4) / (4 + 1) (10 + 5i) / 5 2 + i ANSWER: 2

Simplify. (3 + i)(-2 - i)

(3 + i)(-2 - i) -6 - 3i - 2i - i^2 -6 - 5i - -1 = -5 - 5i

Simplify. (4 + 2i)^2

(4 + 2i)^2 (4 + 2i)(4 + 2i) 16 + 8i + 8k + 4i^2 18 + 16i + 4(-1) 18 + 16i - 4 = 12 + 16i

Solve. (50Σk=1)(2k-1) = ?

(50Σk=1)(2k-1) 2(50) - 1 = 100 - 1 = 99 50[(1 + 99) / 2)] 50 (100 / 2) 50(50) = 2500

Simplify. (a^4)(b^2)[a^(-1/2)][b^(-3)]

(a^4)(b^2)[a^(-1/2)][b^(-3)] (a^3.5)[b^(-1)] = a^3.5 / b

Simplify. (a^b)^c

(a^b)^c a^(bc)

Binomial Coefficient Formula

(n choose k) = (n!) / [(n - k)!(k!)]

Given "n" objects, the number of groups of size "k" when order does NOT matter is . . . .

(n choose k) = n! / [(n - k)!(k!)]

Geometric Series Format

(nΣk)(r^k) for some r < 1

What will a problem look like that indicates you should use the binomial theorem?

(x + y)^n

Expand. (x-7)(2x+3)

(x-7)(2x+3) 2x^2 + 3x -14x - 21 = 2x^2 -11x - 21

Expand. (x^2 - 7x +1)(x^2 + 2x + 2)

(x^2 - 7x +1)(x^2 + 2x + 2) x^4 + 2x^3 + 2x^2 - 7x^3 - 14x^2 - 14x + x^2 + 2x + 2 = x^4 - 5x^3 - 11x^2 - 12x + 2

Simplify. (x^4)(y^4) = ?

(xy)^4

Solve. (∞Σk=1)(1/3)^k

(∞Σk=1)(1/3)^k (1/3) / [1 - (1/3)] (1/3) / (2/3) 3/6 = 1/2

Solve. (∞Σk=1)2(7/8)^k

(∞Σk=1)2(7/8)^k 2(∞Σk=1)(7/8)^k 2[(7/8) / (1 - (7/8))] 2[(7/8) / (1/8)] 2(7) = 14

Linear inequalities refer to all points on ___ side of a line either __________ (≤, ≥) or __________ (<, >) the line itself.

- 1 -including -excluding

In exponential functions with a negative exponent, as x becomes negative it approaches ___ , and as it becomes positive it approaches ___ .

- ∞ -0

In exponential functions, as x becomes negative, it approaches ___ , and as x becomes positive, it approaches ___ .

-0 -∞

Complex Numbers Property i^2 = ?

-1

Solve for x. -2x + 3 = 9x - 11

-2x + 3 = 9x - 11 11x = 14 x = 14/11

Solve. -2x + 5 > 0

-2x + 5 > 0 -2x > -5 x < 5/2

Irrational Numbers w/ Classic Examples

-A subset of real numbers -Are all of the real numbers that are not rational -Have never ending repeating decimals -Ex: π; 2√2

Composition w/ Format Example

-Applying one function, then the other -Format Example: f(g(x))

Rational Numbers

-Are a subset of real numbers -Are ratios / fractions of integers -Any number in the form of "p/q", when q does not equal 0

Integers

-Are a subset of real numbers -Numbers without decimal or fraction parts -Can be positive or negative

What is the domain and range of . . . . f(x) = 1/x

-Domain: (-∞,0) U (0,∞) -Range: (-∞,0) U (0,∞)

What is the domain and range of . . . . f(x) = x^2

-Domain: (-∞,∞) -Range [0,∞)

What is the domain and range of . . . . f(x) = a^x

-Domain: (-∞,∞) -Range: (0,∞)

What is the domain and range of . . . . f(x) - I x I

-Domain: (-∞,∞) -Range: [0,∞)

What is the domain and range of . . . . f(x) = log (x)

-Domain: (0,∞) -Range: (-∞,∞)

What is the domain and range of . . . . f(x) = √x

-Domain: [0,∞) -Range: [0,∞)

What is the difference between an exponential and power function?

-Exponential: Variable is in the exponent -Power: Number is in the exponent

Complex Numbers

-Extend the real numbers by introducing the unit "i", which equals √-1

What are the ways you can find the roots of quadratics / second degree polynomials?

-Factoring -Quadratic Formula

What are different methods of solving a quadratic equation?

-Factoring -Using the quadratic formula -Completing the square

Inverse Functions w/ Format Notation

-Functions that undo each other -f^(-1)(x)

How fast do exponential functions grow and decay?

-Grow: Double -Decay: Half

What are the rules to tell, using the discriminant of a quadratic formula, how many solutions a quadratic will have?

-If this value is positive, there are 2 solutions (if this is the case, the parabola touches the x-axis twice) -If this value equals 0, there is 1 solution (if this is the case, the parabola only touches the x-axis once) -If this value is less than 0, there are no solutions

Properties of Logarithms Quotient Property w/ Format Example

-If you have the log "of" 2 numbers divided by each other (aka the log of a fraction), it's the same as the log of the first number minus the log of the second number -Ex: loga (x/y) = loga (x) - loga (y)

Properties of Logarithms Product Property w/ Format Example

-If you have the log "of" 2 numbers multiplied by each other, it's the same as the log of the first number plus the log of the second number -Ex: loga (xy) = loga (x) + loga (y)

Properties of Logarithms Exponential Property w/ Format Example

-If you have the log "of" a number raised to an exponent, you can bring the exponent in front of the log so that it is multiplied by the log -Ex: loga (x^y) = y loga (x)

How can you check to see if a number is part of the domain of a function?

-It is part of the domain as long as it doesn't make the function be dividing by 0 1) Plug it into the denominator 2) Set that equal to 0 3) If it does NOT equal 0, then it IS part of the domain

You can ALWAYS plot a function by plugging in several values for x to get y-values. However, what is the simpler method?

-Make sure to know the general shape of the graph -Plug in a couple values -Connect the dots to complete the line and then extend them in the general direction of the shape of the function OR -Apply the transformation rules for shifting / stretching / shrinking of the function

How many unique order triples of letters exist if each letter may only be used once?

-Order DOES matter because they need to be unique orders of triples of letters n = 26; k = 3 n! / k! = 26! / 3!

10 people are at a dance. How many ways can you choose pairs?

-Order does NOT matter here because a pair will be of the same 2 people, no matter who is chosen first n = 10; k = 2 n! / k! 10! / [(10 - 2)!(2!) 10! / [(8!)(2!)] (10 ⋅ 9) / 2 90 / 2 = 45

Solve. y = 2x + 1 { y = -x - 1

-Plot the equations ⋅2x + 1 = -x - 1 3x = -2 x = -2/3 ⋅y = -(-2/3) - 1 y = (2/3) - 1 y = -1/3 ANSWER: (-2/3 , -1/3)

What are the steps to solving higher order systems of inequalities?

-Plot the lines and then look where they overlap 1) Plot each line (use dotted line with < > and normal line with ≤ ≥) 2) Shade the region of each inequality (shade up for > ≥ and shade down for < ≤) 3) Find the overlap in the shaded regions on the graph

How do you solve a system of equations with an order 1 and order 2 polynomial?

-Put the equations in the format of "y=" -Set them equal to each other -Put this new equation in the format of "y=" -The result will be a quadratic. Thus, solve for x using either factoring, the quadratic formula, or completing the square -Plug this x-value into one of the original "y=" equations -Solve for y -These x and y values are the coordinate pair of intersection

What do systems of equations find? What is their notation?

-The intersection where the 2 lines meet, as written by the shared x,y coordinate -Notation: {

Factorial w/ Notation Example

-The product of an integer and all the integers below it -Notation Example: !

Simiplify. 100! / 99!

-Think logically about this ANSWER: 100

List the steps for how to solve absolute value inequalities?

-Use the number line method 1) Solve the corresponding equation (remember to solve for both the positive and negative) 2) Plot solutions on the number line 3) Check points in the region on the number line to see if they work

What are the steps in determining if an equation or inequality is a function or not?

-Use the vertical line test 1) Graph the function if not already done 2) Check to see if you would draw a vertical line at any place along that function, how many times it would hit the function 3) If it only hits the function once, it is a function, but if it hits the function more than once at any place, then it is not a function

Discriminant of a Quadratic

-Used to determine the number of solutions in a quadratic -Is the value of the numbers underneath the square root sign in the quadratic formula

What is the notation of a conjugate? w/ Example

-Whatever variable the equation contains with a bar over the top of it -Ex: y --> y bar

Odd Symmetry

-When a function is symmetric about the origin -You have to flip the function twice

When factoring, the numbers must __________ to form the middle term and __________ to form the last term of a quadratic.

-add -multiply

Exponential functions have rapid __________ and rapid __________.

-growth -decay

Exponentials have __________ growth because each value __________.

-rapid -doubles

The complex plane has __________ numbers on the x-axis and __________ numbers on the y-axis.

-real -imaginary

Complex numbers appear naturally in __________ and __________ polynomials. This can happen in the __________ formula if the discriminant is negative.

-roots -quadratic -quadratic

If Matrix A is x ⋅ y, it has ___ rows and ___ columns.

-x -y

Constants are degree ___ polynomials.

0

Properties of Logarithms loga (1) = ?

0

Function Example f(1/2) = ?

0 (because it is NOT a whole number / integer)

Function Example f(π) = ?

0 (because it is NOT a whole number / integer)

Any number with 0 as the exponent equals what?

1

Factorial Property 0! = ?

1

Factorial Property If n=k in (n choose k), what does that always equal?

1

Properties of Logarithms logx (x) = ?

1

Function Example f(0) = ?

1 (because it IS a whole number / integer)

Function Example f(3) = ?

1 (because it IS a whole number / integer)

Find the determinant of this matrix. 1 -1 [ ] 0 2

1 -1 det[ ] = (1⋅2) - (-1⋅0) = 2 - 0 = 2 0 2

Simplify. 1 / (i + 1)

1 / (i + 1) [1 / (i + 1)][(-i + 1)/(-i + 1)] (-i + 1) / [(i + 1)(-i + 1)] (-i + 1) / (-i^2 + 1) (-i + 1) / [(-1)^2 + 1] (-i + 1) / (1 + 1) = (-i + 1) / 2

Find the 13th term of (x+y)^27.

1) (27Σk=13) = [n / (n-k)!(k!)]x^(n-k)y^k [27 / (27-13)!(13!)]x^(27-13)y^13 = [27 / (14!)(13)!]x^15y^12 (THIS IS FOR SURE THE CORRECT ANSWER)

What are the steps to solving exponential and logarithmic equations?

1) Change the equation to a log or exponential 2) Apply exponential and logarithmic properties

What are the steps to solving a quadratic inequality?

1) Factor or use the quadratic equation 2) Find the zeroes

What are the steps to factor higher order polynomials?

1) Factor out any common factors 2) Guess and check until you factor the problem correctly

What are the steps in dividing complex numbers?

1) Find the conjugate of the denominator 2) Multiply the original fraction by a convenient 1 composed of that conjugate to clear the denominator 3) Simplify -Make sure the answer does NOT have a complex number in the denominator

What are the steps to solving a problem where they ask you to use the binomial theorem / to expand a polynomial?

1) Look at the exponent (n) 2) There will be 1 more term than that number in the exponent 3) The first term will be with x to that exponent, and y to the 0 exponent (thus 1, and y will not be present in this term) 4) Continue writing the x and y terms with the exponents of the term "x" decreasing by 1 each time down until 0, and the exponents of the term "y" increasing by 1 each time up to the exponent in the original problem 5) To get each of the coefficients, take the exponent in the original problem, and add 1 to it 6) Go to that number row in Pascal's Triangle 7) There, will be listed the coefficients for each of the terms 8) Put plus signs in between each of the terms

How do you simplify radicands?

1) Look at the number under the radical and see if it has a factor that is a perfect square 2) If so, take the square root of that term and put it outside the radical 3) Leave the other number it was multiplied by under the radical 4) If you end up having 2 numbers outside the radical, multiply them

What are the steps to find a specific term in a binomial that you would usually use the binomial theorem for?

1) Make "k" in the binomial theorem equal to the number term you are looking for minus 1 (ex: if you are looking for the 13th term, then k=12) 2) Solve the binomial theorem for only this term by filling in the binomial theorem with these values of n and k 3) When doing the exponents of x and y, count them out by hand

What are the steps to get rid of a negative exponent?

1) Move the term with the exponent to the denominator 2) Make the exponent positive

Properties of Exponents How do you multiply 2 terms with different bases but the same exponents?

1) Multiply the bases 2) Raise this to the exponent

Solve. y ≥ x + 1 { y ≥ 3x - 1

1) Plot each line on the same coordinate plane with a normal line 2) Shade upwards because it is ≥ 3) Shade over where the shaded regions overlap on the graph (THIS IS THE ANSWER)

What are the steps to graph an inequality?

1) Put it into slope intercept form 2) Plot that equation on the graph 3) Draw it as a dotted line if the inequality is <>, and draw it as a normal line if the inequality is ≤ ≥ 4) Shade the corresponding side (> ≥ shade up) (< ≤ shade down) 5) To check if you shaded the correct side of the line, test the point (0,0) and see if the answer lies on the corresponding side as in the inequality

What are the steps to graphing a piecewise function?

1) Put the endpoints of each interval into the function to get a coordinate(s) 2) Plot the points on a coordinate plane 3) Connect the dots of the same interval coordinates to create lines, if needed

What are the steps in testing to see if a function has odd symmetry?

1) Set f(x) = -f(x) 2) Plug in a test value for x 3) Simplify 4) If both sides equal each other, then it has odd symmetry

What are the steps in testing to see if a function has even symmetry?

1) Set f(x)=f(-x) 2) Plug in a test value for x 3) Simplify 4) If both sides equal each other, then it has even symmetry

What are the steps to find the zeroes of a quadratic inequality with a < ≤ sign?

1) Set up the factored parts to "A < ≤ 0" or "B > ≥ 0" AND "A > ≥ 0" or "B < ≤ 0" 2) Write the values of x

What are the steps to finding the zeroes of a quadratic inequality with a > ≥ sign?

1) Set up the factored parts to "A > ≥ 0" or "B > ≥ 0" AND "A ≤ < 0" or "B ≤ < 0" 2) Take 1 value from each of these to summarize them both

What are the steps to add or subtract matrices?

1) Simply add or subtract the corresponding terms of the matrix

What are the steps to multiply a matrix by a scalar?

1) Simply multiply the scalar to the corresponding terms of the matrix

What are the steps in solving an absolute value equation?

1) Solve "ax + b = c" and "ax + b = -c"

What are the steps in solving a system of equations?

1) Solve for y in the equations so that both of the equations are in the form of "y=" 2) Plot both of the lines on the same coordinate plane to have a general idea of where they will intersect 3) Set the equations equal to each other 4) Solve for x 5) Plug this x-value into one of the "y=" equations 6) Solve for y 7) These x and y values are the coordinate pair where the lines intersect

What are the steps for the easiest way to write an inverse function.

1) Switch x and y 2) Solve for y

What is the easiest way to write the original function, given an inverse function?

1) Switch x and y 2) Solve for y

Simplify. [(-2)/((x^2)-1)] + [1/(x+1)] - [1/(x-1)]

[(-2)/((x^2)-1)] + [1/(x+1)] - [1/(x-1)] (-2+x-1-x-1) / [(x^2)-1] = -4 / [(x^2)-1]

Transform this into an exponential function. log2 (16) = 4

2^4 = 16

Solve. 2^x = 4^(2x-1)

2^x = 4^(2x-1) log2 (2^x) = log2 (4^(2x-1)) x = (2x-1)log2 (4) x = (2x-1)2 x = 4x - 2 3x = 2 x = 2/3

In order to have a determinant, a matrix must be a ___x___ matrix.

2x2

Solve. 2x^2 + 3x = 4

2x^2 + 3x = 4 2x^2 + 3x - 4 = 0 x = [-3 ± √((3^2) - 4 ⋅ 2 ⋅ -4)] / (2 ⋅ 2) x = [-3 ± √(41)] / 4

Solve. 3^((x^2)-1) = 1

3^((x^2)-1) = 1 log3 (3^((x^2)-1) = log3 (1) ((x^2)-1)log3 (3) = log3 (1) (x^2) - 1 = 0 (x + 1)(x - 1) = 0 x = -1; 1

Solve. 3^((x^2)-3x) = 1/9

3^((x^2)-3x) = 1/9 log3 (3^((x^2)-3x)) = log3 (1/9) ((x^2)-3x)log3 (3) = log3 (1/9) x^2 - 3x = -2 x^2 - 3x + 2 = 0 (x - 2)(x - 1) = 0 x = 2; 1

Simplify. 3^[log3 (4)]

3^[log3 (4)] log3 (3^4) <-- optional step = 4

Simplify. 3^x ⋅ 2^x ⋅ 6^(-x)

3^x ⋅ 2^x ⋅ 6^(-x) 6^x ⋅ 6^(-x) 6^(x-x) = 6

Solve. 3x + 1 ≥ 2

3x + 1 ≥ 2 3x ≥ 1 x ≥ 1/3

Find all solutions to 3x^2 - 4x = -4.

3x^2 - 4x = -4 x^2 - (4/3)x = (-4/3) x^2 - (4/3)x + (4/9) = (-4/3) + (4/9) x^2 - (4/3)x + (4/9) = (-12/9) + (4/9) x^2 - (4/3)x + (4/9) = -8/9 [x - (2/3)]^2 = -8/9 x - (2/3) = ± √(-8/9) x - (2/3) = ± [√(8/9)](i) x = (2/3) ± (1/3)(√8)i

Solve for y. 3y - 4x + 7 = 2y + 8x + 1

3y - 4x + 7 = 2y + 8x + 1 y = 12x - 6

Factor. 4x^2 + 12x + 9

4x^2 + 12x + 9 = (2x+3)^2

Solve. 5 choose 4

5! / [(5 - 4)!(4!) 120 / [1(24)] = 5

Solve for y. 5x - 6y - 2 = 12x + y + 2

5x - 6y - 2 = 12x + y + 2 -7y = 7x + 4 y = -x - 4/7

Solve. (7 choose 4)

7! / [(7-4)!(4)!] 7! / (3!)(4!) (7⋅6⋅5⋅4⋅3⋅2⋅1) / (3⋅2⋅1⋅4⋅3⋅2⋅1) 7 ⋅ 5 = 35

Inverse Function Property [f^(-1)(x) o f)](x) = ? = ?

= (f o f^-1)(x) = x

Arithmetic Series {a(subk)}(nOFk=1) = ?

= (nΣk=1)[a(subk)] =

Sequence

A list of numbers in a given order

Vector

A matrix with only 1 row or column

Scalar

A number that is being multiplied by a matrix

Arithmetic Series

A series where elements change by a fixed amount

Conjugate

A term that is composed by switching the sign of the imaginary component of a complex number

Write a function to represent the area of a rectangle, given that A=lw

A(l,w) = lw

What is a helpful way to solve factorials?

Actually write out all of the terms of the factorial so that you can easily see cancellations

Properties of Exponents How do you multiply 2 terms with the same base but different exponents?

Add the exponents

How do you solve a linear equation

Compute values of x given y, and values of y given x

On the CLEP exam, if asked to factor a problem, what is a short cut I can do?

Expand the multiple choice answers until I get the polynomial in the question

How do you expand polynomials?

FOIL

What is another way of saying, "Find the zeroes of the polynomial"?

Find the roots of the polynomaial

What do you need to do to add / subtract fractions?

Get a common denominator

Solve. I 2x + 1 I = 4

I 2x + 1 I = 4 ⋅2x + 1 = 4 2x = 3 ANSWER: x = 2/3 ⋅2x + 1 = -4 2x = -5 ANSWER: x = -5/2

Solve. I 3x + 4 I = 1

I 3x + 4 I = 1 ⋅3x + 4 = 1 3x = -3 ANSWER: x = -1 ⋅3x + 4 = -1 3x = -5 ANSWER: x = -5/3

Solve. I 6x-1 I ≤ 2

I 6x-1 I ≤ 2 ⋅6x - 1 = 2 x = 1/2 ⋅6x - 1 = -2 x = -1/6 -Plot these values on a number line TEST VALUES: -1; 0; 1 (plug back into absolute value inequality) -Value of 0 is true ANSWER: [-1/6 , 1/2]

What is the format for absolute value inequalities?

I ax+b I <, >, ≤, ≥ c

What is the format of absolute value equations?

I ax+b I = c

Solve. I x-7 I ≥ 4

I x-7 I ≥ 4 ⋅x - 7 = 4 x = 11 ⋅x - 7 = -4 x = 3 -Plot these values on a number line TEST VALUES: 0; 10; 20 (plug back into original absolute value inequality) -Values of 0 and 20 are true ANSWERS: (-∞,3] U [11,∞)

Given an equation or inequality, how can you tell if 2 lines are perpendicular?

If the slopes are opposite reciprocals of each other

Given an equation or inequality, how can you tell if 2 lines are parallel?

If they have the same slope

Compute. [(3x-1) / (x^2-2)] + [1/(x-1)]

[(3x-1) / (x^2-2)] + [1/(x-1)] [(3x-1)(x+1) + (x^2-2)] / [(x^2-2)(x-1) (3x^2 - 3x - x + 1 + x^2 - 2) / (x^3 - x^2 - 2x + 2) = (3x^2 - 4x + 2) / x^3 - x^2 - 2x + 2)

Simplify. [(4√9)^8] / [(4^2)-(3√8)]

[(4√9)^8] / [(4^2)-(3√8)] [(9^(1/4))^8] / (16-2) [9^((1/4)(8))] / 14 (9^2) / 14 = 81 / 14

Simplify. [(a^4)(b^3)] / [(a^3)(b^3)]

[(a^4)(b^3)] / [(a^3)(b^3)] = a

Simplify. [(x^2 ⋅ x^(-3)) / x^5] (x^4/x) (x)

[(x^2 ⋅ x^(-3)) / x^5] (x^4/x) (x) [x^(-1) / x^5] (x^4/x) (x) [x^(-1) / x^5] ⋅ x^4 = x^3 / x^5

How do you multiply fractions?

Multiply numerators and multiply denominators

How do you divide by a fraction / what do you do if you have a fraction in the denominator?

Multiply the entire term by the reciprocal fraction

Properties of Exponents How do you solve a term with an exponent, and that whole term is raised to another exponent?

Multiply the exponents

Real Numbers

Numbers that does not have an imaginary component

A colony of bacteria starts with 10 members. Every hour, the amount of bacteria doubles. Write the population P(t), where t=time / rate of growth (hrs).

P(t) = 10(2^t) t^2 is the doubling property

What is the more common name for a second degree polynomial?

Quadratic

Is the number 0 a real number, an integer, or a complex number?

Real number

What does this transformation do? f(x) --> -f(x)

Reflects the function over the x-axis

What does this transformation do? f(x) --> f(-x)

Reflects the function over the y-axis

What does this transformation do? f(x) --> f(x)-a

Shifts the function "a" points downward

What does this transformation do? f(x) --> f(x+a)

Shifts the function "a" points to the left

What does this transformation do? f(x) --> f(x-a)

Shifts the function "a" points to the right

What does this transformation do? f(x) --> f(x)+a

Shifts the function "a" points upward

What is the formula for finding the determinant of a 2x2 matrix?

a b det[ ] = ad - bc c d aka 1) Multiply the diagonals 2) Take ad minus bc

Properties of Exponents How do you divide terms with exponents?

Subtract the exponents

Series

Sums of sequences

What do you need to remember to do when multiplying or dividing by a negative in an inequality?

Switch the direction of the inequality sign

Degree of Polynomial

The highest exponent of x in the polynomial

What is the function simplified down to when you have a number raised to a log with the bases of the exponential term and log the same?

The remaining number / the "of #" in the log

Domain

The set of allowable inputs (x-values)

Range

The set of allowable outputs (y-values)

Geometric Series

The sum of terms where r is less than 1

T or F Functions can be added, subtracted, multiplied, and divided.

True

T or F It is possible for a system of equations to be of different orders, but you can still use the same techniques as solving a system of equations with equations of the same order.

True

T or F Roots and powers undo each other.

True

Solve. (10 choose 10)

Use factorial property = 1

Even Symmetry

When a function is symmetric across the y-axis

A series is written in what format?

a(sub1) + a(sub2) + a(sub3) + . . . + a(subn) = (nΣk=1)[a(subk)]

A sequence is written in what format?

a(sub1), a(sub2), a(sub3), . . . a(subn) = {a(subk)}(n of k) -When k=1, that answer equals a(sub1) -When k=2, that answer equals a(sub2) -Etc.

Logarithms as Inverse of Exponentials loga (a^x) = ?

a^[loga (x)] = x

Simplify. (a^x)^y = ?

a^xy

What is the format of quadratic inequalities?

ax^2 + bx + c <, >, ≤, ≥ 0

What is the format for second degree polynomials?

ax^2 + bx + c = 0

Simplify. i(i + 1)

i(i + 1) i^2 + i = -1 + i

What is the most important rule to keep in mind when multiply complex numbers and simplifying?

i^2 = -1

Logarithms and exponentials are __________ of each other.

inverses

Simplify. log10 (x^2 ⋅ y^3)

log10 (x^2 ⋅ y^3) log10 (x^2) + log10 (y^3) = [2 log10 (x)] + [3 log10 (y)]

Simplify. log10 [(x^2)/x^4) - x^(-2) + 1]

log10 [(x^2)/x^4) - x^(-2) + 1] log10 [x^(-2) - x^(-2) + 1] log10 (1) = 0

Simplify. log2 (16)

log2 (16) log2 (2^4) 4 log2 (2) = 4

Transform this into a logarithmic function. 2^4 = 16

log2 (16) = 4

Simplify. log2 [(x^2)/8^y)]

log2 [(x^2)/8^y)] [log2 (x^2)] - [log2 (8^y)] [2 log2 (x)] - [y log2 (8)] = [2 log2 (x)] - 3y

Simplify. log4 [(2^x)(2^x)]

log4 [(2^x)(2^x)] log4 (2⋅2)^x log4 (4^x) = x

Solve. log5 (x+2) = 2

log5 (x+2) = 2 5^2 = x + 2 x + 2 = 25 x = 23

If you have variables in the exponent (thus, an exponential equation) then put it into __________ form and solve it that way.

logarithmic

Factorial Property ("n" choose "n-1") = ?

n

Given "n" objects, the number of groups of size "k" when order DOES matter is . . . .

n! / k!

Solve. I 3x + 4 I = -1

no solutions (because the answer to the equation is negative)

The absolute value of a number is always __________ unless there is a negative sign outside of the absolute value bars.

positive

The answer to an absolute value equation, given that there is no negative sign outside the absolute value bars, is always __________.

positive

Most quantities used to describe things in the real world are __________ numbers.

real

Logarithms grow very __________.

slowly

Matrices can be added and subtracted ONLY if . . . .

they are the same size / have the same number of rows and columns (CANNOT HAVE # OF ROWS AND COLUMNS FLIP FLOPPED)

loga __________ a^x and vise versa

undoes

(√x)^2 = ?

x

Simplify. 5^[log5 (x)]

x

f(x) = 0 if . . . .

x is NOT a whole number

f(x) = 1 if . . . .

x is a whole number / integer

Expand. x[(x-2)^2](x+4)

x[(x-2)^2](x+4) x(x-2)(x-2)(x+4) x(x^2 - 4x + 4)(x+4) x(x^3 + 4x^2 - 4x^2 - 16x + 4x^2 + 16x x^4 + 4x^3 - 4x^3 - 16x^2 + 4x^2 + 16x = x^4 - 12x^2 + 16x

Simplify. x^(1-y) ⋅ x^(y+7) ⋅ x^(-6)

x^(1-y) ⋅ x^(y+7) ⋅ x^(-6) x^(1-y+y+7-6) = x^2

What is another way to write √x ?

x^(1/2)

Solve. y = x^2 + 2x + 1 { y = -2x + 3

x^2 + 2x + 1 = -2x + 3 x^2 + 4x - 2 = 0 x^2 + 4x = 2 ⋅x^2 + 4x + 4 = 6 (x + 2)^2 = 6 x + 2 = ± √6 x = -2 ± √6 ⋅y = -2(-2 ± √6) + 3 y = 4 ± -2√6 + 3 y = 7 ± -2√6 ANSWER: (-2 ± √6 , 7 ± -2√6)

Solve. x^2 + 3x - 4 < 0

x^2 + 3x - 4 < 0 (x + 4)(x - 1) < 0 ⋅x + 4 < 0 --> x < -4 OR x - 1 > 0 --> x > 1 (impossible) ⋅x + 4 > 0 --> x > -4 OR x - 1 < 0 --> x < 1 ANSWER: x > -4 OR x < 1 (-4,1)

Solve. x^2 + 5x + 4 ≤ 0

x^2 + 5x + 4 ≤ 0 (x + 4)(x + 1) ≤ 0 ⋅x + 4 ≤ 0 --> x ≤ -4 OR x + 1 ≥ 0 --> x ≥ -1 (impossible) ⋅x + 4 ≥ 0 --> x ≥ -4 OR x + 1 ≤ 0 --> x ≤ -1 ANSWER: x ≥ -4; x ≤ -1

Factor. x^2 + 6x + 8

x^2 + 6x + 8 = (x+4)(x+2)

Find the roots of the following equation: x^2 + 7x + 9

x^2 + 7x + 9 x = [(-7) ± √((7^2)-(4⋅1⋅9)] / (2⋅1) x = [(-7) ± √(49-36)] / 2 x = [(-7) ± √13] / 2

Solve. x^2 - 1 > 0

x^2 - 1 > 0 (x + 1)(x - 1) > 0 ⋅x + 1 > 0 --> x > -1 OR x - 1 > 0 --> x > 1 ⋅x + 1 < 0 --> x < -1 OR x - 1 < 0 --> x < 1 ANSWER: x > -1 OR x < 1 (-1,1)

Factor and find the zeroes. x^2 - 3x - 10 = 0

x^2 - 3x - 10 = 0 (x - 5)(x+2) = 0 x = 5; -2

Factor. x^2 - 4

x^2 - 4 = (x+2)(x-2)

Factor and find the zeroes. x^2 - 4x + 4 = 0

x^2 - 4x + 4 = 0 (x-2)^2 = 0 x = 2

Factor. x^2 - 6x + 9

x^2 - 6x + 9 (x-3)(x-3) = (x-3)^2

Factor and find the zeroes. x^2 - 7x + 12 = 0

x^2 - 7x + 12 = 0 (x-4)(x-3) = 0 x = 4; 3

Solve. x^2 - x + 2 = 0

x^2 - x + 2 = 0 x = [1 ± √((-1^2) - 4 ⋅ 1 ⋅ 2)] / (2 ⋅ 1) x = [1 ± √(1-8)] / 2 x = [1 ± √(-7)] / 2 x = [1 ± √(7) ⋅ i] / 2

Find the roots / zeroes of: x^3 - 9x

x^3 - 9x x(x^2 - 9) x(x+3)(x-3) x=0; -3; 3

Factor. x^3 - x

x^3 - x x(x^2 - 1) = x(x+1)(x-1)

Simplify. (x^2)(x^3) = ?

x^5

Expand. (x + y)^5

x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

What is the format for a linear inequality?

y <, >, ≤, ≥ mx + b

Solve for x when y = 2. y = 3x - 1

y = 3x - 1 2 = 3x - 1 3x = 3 x = 1

What is the format for first degree polynomials?

y = ax + b

What is the format of a linear equation?

y = ax + b

What is the format of a quadratic equation?

y = ax^2 + bx + c

Simplify. y = log10 (100)

y = log10 (100) 10^y = 100 2

What is the formula of a logarithmic function?

y = loga (x)

What is the format for slope intercept form?

y = mx + b

What is the formula of an exponential function?

y = x^x

Complex Numbers Property i = ?

√(-1)

Simplify. √3 ⋅ √27

√3 ⋅ √27 √3 ⋅ 3√3 3√9 3 ⋅ 3 = 9

Solve. 3y - 4 = 6x + 1 { 2x = y - 7

⋅3y - 4 = 6x + 1 3y = 6x + 5 y = 2x + 5/3 ⋅2x = y - 7 y = 2x + 7 -Plot the equations -These lines have the same slopes, thus they are parallel, and there is no solution because they do not intersect