Solving Logarithmic and Exponential Equations, Properties of Logarithms, Graphing Logarithmic Functions, Exponential Functions, Inverse Functions, Composite Functions, Rational and Polynomial Inequalities, Graphing Rational Functions, Rational Functi...
The range of the inverse of f(x) = x³
( -∞ , ∞ )
(x+7) /(2x+1) > 2
(-1/2, 5/3)
(2-x) / [(x-5)(x+10)] ≤ 0
(-10, 2] ∪ (5, ∞)
(x+68)/(x+8)≥5
(-8,7]
(x²-9)/(x+5)<0
(-inf, -5)U[-3,3]
(x+32)/(x+6)≤3
(-inf, -6]U[7, inf)
(x + 9) (x - 2) (x + 5) < 0
(-∞, -9) ∪ (-5, 2)
5x² + 20x + 30 > 3x² + 4x
(-∞,-5) ∪ (-3,∞)
-2
-10 + log₃ (x + 3) = -10
2
-10 + log₃ (x + 3) = -10
Steps to solve algebraically for the inverse of the function
1. Replace f(x) with y in the equation for f(x) 2. Interchange x and y 3. Solve for y. 4. Replace y with f⁻¹(x)
(2b²-7b+4)÷(b-3)
2b-1 + (1/b-3)
32
2log₄ x = 5
(6x²+13x+6)÷(3x+2)
2x+3
100
3log x = 6
(9x²-42x+45)÷(3x-8)
3x-6 - (3/3x-8)
6
3ⁿ⁻² = 81
1
4ⁿ⁺² = 64
The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as mc023-1.jpg, where mc023-2.jpg and is the least intense sound a human ear can hear. What is the approximate loudness of the dinner conversation, with a sound intensity of 10-7, Rajah has with his parents?
50 Db
(7y²-3y-4)÷(y-1)
7y+4
For every number x in its domain, the number -x is also in the domain and f(-x) = f(x)
A function is even if
Its graph is symmetric with respect to the y-axis
A function is even if and only if
For every number x in its domain, the number -x is also in the domain and f(-x) = -f(x)
A function is odd if
Its graph is symmetric with respect to the origin
A function is odd if and only if
Exponent
A number placed above and to the right of another number to show that it has been raised to a powe
One-to-one Function
A property of functions where the same value for y is never paired with two different values of x (the function passes the horizontal line test)
Horizontal Line Test
A way to establish if a function is one-to-one when looking at a function's graph.
Vertical Line Test
A way to establish that a relation is a function.
Reflection about the line y = x.
A way to graphically see if two functions are inverses of each other.
y=|x| Absolute value represents the distance an integer is from zero.
Absolute Value Function
If there is a number u in I for which f(x) < f(u) for all x in I, then f(u) is the absolute maximum of f on I and we say the absolute value of f occurs at u (less than or equal to)
Absolute maximum of f
If there is a number v in I for which f(x) > f(v) for all x in I, then f(v) is the absolute minimum of f on I and we say the absolute minimum of f occurs at v. (greater than or equal to)
Absolute minimum of f
If a and b, where a is not equal to b, are in the domain of a function y = f(x), the average rate of change of f from a to b is defined as ^y/^x = f(b) - f(a) / b -a where b does not equal a (^ are triangles)
Average rate of f
What is mc017-1.jpg written as a single logarithm?
B) mc017-3.jpg
The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as mc026-1.jpg, where mc026-2.jpg and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is 10-10. Ahmad, Brandon's neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is 10-4. How does Brandon's sound intensity level compare to Ahmad's?
Brandon's sound intensity is mc026-3.jpg the level of Ahmad's.
Which expression is equivalent to mc009-1.jpg?
C) mc009-4.jpg
Which expression is equivalent to mc014-1.jpg?
C) mc014-4.jpg
A type of step function. It rounds the input (x) up to the greatest integer .
Ceiling Function
A number that is multiplied by a variable(s)
Coefficient
True! f[g(x)] is generally not equal to g[f(x)]. Consider f(x) = 2x, and g(x) = x - 3 f[g(x)] = 2(x - 3) = 2x - 6 g[f(x)] = (2x) - 3 = 2x - 3 f[g(x)] is not equal to g[f(x)].
Composition of functions is not commutative. True or false?
A fixed value that does not appear with a variable(s)
Constant
A function f is constant on an open interval I if, for all choices of x in I, the values of f(x) are equal
Constant
When "y" equals a constant number (no x-variable).
Constant Function
A ratio when the dependent, y-value, changes at a constant rate for each independent, x-value.
Constant Rate of Change
A constant ratio between two proportional quantities denoted by the symbol k
Constant of Proportionality
x-axis : angles (degrees or radians) y-axis : adjacent (x) side of a right triangle
Cosine Function
A number raised to the 3rd power.
Cubic Function
A number that when multiplied three times equals the given number.
Cubic Root Function
Which is the graph of a logarithmic function?
D
What is mc007-1.jpg rewritten using the power property?
D) mc007-5.jpg
A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) >f(x2)
Decreasing
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Distance Formula
The set of all real numbers except the values that are zeros of the denominator
Domain of rational functions
This happens when the graph as x approaches positive infinity (+∞) or negative infinity (−∞)
End Behavior
The x-axis is an asymptote.
Exponential Function
If f is a continuous function whose domain is a closed interval [a,b], then f has an absolute maximum and an absolute minimum [a,b]
Extreme value theorem
The absolute maximum and absolute minimum of a function f are sometimes called the extreme values of f on I
Extreme values
The FACTOR THEOREM links factors and zeros of a polynomial. It is commonly applied to factorizing and finding the roots of polynomial equations. The theorem states that is a factor of a polynomial 'f(x)' if 'r' is a root of 'f(x)'. Copy and paste the following link into your browser to learn more about using factor theorem with polynomial functions https://youtu.be/-yon7Abs3PY
Factor Theorem
Set cancelled parts of the fraction = 0
Find holes
Look at degrees of numerator and denominator
Find horizontal asymptotes
Set denominator = 0 (after factoring and canceling)
Find vertical asymptotes
Set numerator = 0 (after factoring and canceling)
Find x-intercept
Plug in 0 for x
Find y-intercept
A type of step function. It rounds the input (x) down to the greatest integer .
Floor Function
Polynomial functions of the form f(x) = xⁿ (where 'n' is either a positive or negative integer) create ONE OF TWO BASIC GRAPHS: Copy and paste the following link into your browser to view examples of graphing polynomial functions of the form f(x) = xⁿ, where 'n' is a POSITIVE integer; as well as the form f(x) = xⁿ, where 'n' is a NEGATIVE integer: https://www.cliffsnotes.com/assets/255809.png Each graph has the origin as its only x‐intercept and y‐intercept. Each graph contains the ordered pair (1, 1). If a polynomial function can be factored, its x‐intercepts can be immediately found. Then a study is made as to what happens between these intercepts, to the left of the far left intercept and to the right of the far right intercept.
Graphing Polynomial Functions
This is an y value that a graph will approach but never touch; where the function is undefined.
Horizontal Asymptote
Multiply the inside of a function my -1
Horizontal flip
Given two functions f[g(x)], step1: substitute the inner function g(x), for x step2: insert into outer function f(x) step3: perform operations step4: combine like terms
How do you compose two functions?
Step1: find the domain (restrictions) of the inner function Step2: combine the functions step3: find the domain (restrictions) of the composite function step4: compose domains.
How do you find the domain of a composite function?
The function y=log(x) is translated 1 unit right and 2 units down. Which is the graph of the translated function?
IT IS NOT C.
If you are given the graph of g(x)=log of 2x, how could you graph f(x)=log of 2x+5?
IT IS NOT Translate each point of the graph of g(x) 5 units left.
What is the range of y=log of 2(x-6)?
IT IS NOT all real number greater than 6
What is the range of y=log of 8x?
IT IS NOT all real numbers not equal to 0.
Which of the following is true about the base b of a logarithmic function?
IT IS NOT b<0 and b DOEST NOT EQUAL TO 1.
Which of the following is the inverse of y=3^x?
IT IS NOT y=log of 1/3x.
Inverse Function
If a function is named f, this can be written as f⁻¹
Horizontal asymptote is y = 0
If degree of denominator is greater than that of the numerator
Step 1: Substitute g(x) for x f[g(x)] = f[x - 8]
If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the first step?
Step 4: combine like terms = 3x² - 48x + 198.
If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the fourth step?
Step 2: Insert into f(x) f[g(x)] = 3(x - 8)² + 6
If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the second step?
Step 3: Factor = 3(x² - 16x + 64) + 6
If you were to evaluate the composite function f[g(x)] for f(x) = 3x² + 6 and g(x) = x - 8, what is the third step?
A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2)
Increasing
Straight line
Linear
y = x
Linear (Identity) Function
A relationship with a constant rate of change represented by a graph that forms a straight line.
Linear Relationship
A function f has a local maximum at c if there is an open interval I containing c so that for all x in I, f(x) < f(c). We call f(c) a local maximum value of f. (less than or equal to)
Local maximum
A function f has a local minimum at c if there is an open interval I containing c so that, for all x in I, f(x) > f(c). We call f(c) a minimum of f. (greater than or equal to)
Local minimum
The y-axis is an asymptote. Inverse of an exponential function.
Logarithmic Function
Which expression is equivalent to log3(x + 4)?
NOT A Probably C
Given mc006-1.jpg and mc006-2.jpg, what is mc006-3.jpg?
NOT B
Given mc003-1.jpg and mc003-2.jpg, what is mc003-3.jpg?
NOT D
Which of the following is equivalent to mc002-1.jpg?
NOT D Probably A
f(x) = e^x Euler's Number e ≈ 2.718281...
Natural Base Function
f(x) = ln x Inverse of natural base function.
Natural Log Function
(x+2)² < 0
No solution
The relationship between two ratios with a rate or ratio that is not constant.
Non-proportional
They only occur in rational functions in which the degree of the numerator is one greater than the degree of the denominator. It is sometimes called a slant asmptote.
Oblique Asymptote
Domain Restriction
Omitting specific values from a relation's set of input values, commonly to ensure that a function's inverse is also a function.
A function with two or more equations each with a specific domain.
Piecewise Function
POLYNOMIAL FUNCTION is any function of the following format: P(x) = a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² + ... + a𝚗-₁x + a𝗇 where the coefficients a₀, a₁, a₂, and so on, are real numbers and n is a whole number. Polynomial functions are evaluated by replacing the variable with a value. The instruction "evaluate the polynomial function P( x) when x is replaced with 4" is written as "find P(4)." For example: If P(x) = 3x³ - 2x² + 5x + 3, then find P(-4). [1] P(-4) = 3(-4)³ - 2(-4)² + 5(-4) + 3 [2] = 3(-64) - 2(16) - 20 + 3 [3] = -192 - 32 - 20 + 3 [4] = -242
Polynomial Function
Having a constant relation in degree or number.
Proportional
A number multiplied by itself.
Quadratic (Squaring) Function
y= ax^2 + bx+ c
Quadratic Function Standard Form
It is the set of all output values.
Range of rational functions
y=1/x, has an asymptote at zero (because the function is undefined if we divide by 0).
Rational (Reciprocal) Function
A rational function is a function in the form R(x)=p(x)/q(x), where p and q are polynomial functions and q is not the zero polynomial.
Rational Function
The set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0.
Rational Numbers
The RATIONAL ZEROS THEOREM (also called the rational root theorem) is used to check whether a polynomial has rational roots (zeros). It provides a list of all possible rational roots of the polynomial equation, where all coefficients are integers. Copy and paste the following link into your browser to learn more about using the rational zeros theorem with polynomial functions: https://youtu.be/NztttLiJnWU
Rational Zeros Theorem
Multiplicative inverse of b is 1/b.
Reciprocal
REMAINDER THEOREM specifies if a polynomial P(x) is divided by (x - r), then the remainder of this division is the same as evaluating P(r), and evaluating P(r) for some polynomial P(x) is the same as finding the remainder of P(x) divided by (x - r). For example: Find P(-3) if P(x) = 7x⁵ - 4x³ + 2x - 11. There are two methods of finding P(-3). ✔︎ Method 1: Directly replace -3 for x. ✔︎ Method 2: Find the remainder of P(x) divided by [ x - (-3)] Copy and paste the following link into your browser to learn more about using remainder theorem with polynomial functions: https://youtu.be/7nWiCQPtMbM
Remainder Theorem
The line containing the points (a, f(a)) and (b, f(b))
Secant line
The numerator and denominator have no common factors other than positive or negative one.
Simplified Form
x-axis : angles (degrees or radians) y-axis : opposite (y) side of a right triangle
Sine Function
The degree of the numerator is more than the degree of the denominator
Slant asymptotes occur when
msec = f(b) - f(a) / b -a = f(a+h) - f(a) / h
Slope of the secant line
A number that when multiplied two times equals the given number.
Square Root Function
x=-b/2a
The axis of symmetry is the vertical line (standard form)
parabola
The graph of a quadratic function is a __
Which statement is true?
The graph of y=log of b(x) +4 is the graph of y=log of b (x) translated 4 units up.
the vertex
The highest or lowest point of a porabola
Domain
The set of all input values of a relation.
Range
The set of all output values of a relation.
axis of symmetry
The vertical line through the vertex
The symmetry that exists for the graph of the function
The words even and odd when applied to a function f describe
The average rate of change of a function f from a to b equals the slope of the secant line containing the two points (a,f(a)) and (b,f(b)) on its graph
Theorem of a Slope of the secant line
It is often helpful to know certain properties that the function that the function has and the impact of these properties on the way that graph will look
To obtain the graph of the function y= f(x)
subtract the outside of a function
Translation downward
Add to the inside of a function
Translation left
Subtract from the inside of a function
Translation right
add to the outside of a function
Translation upwards
A ratio between two different units where one of the terms is 1.
Unit Rate
This is an x value that a graph will approach but never touch; where the function is undefined.
Vertical Asymptote
Multiply the outside of a function by -1
Vertical flip (reflection)
Holes and vertical asymptotes
What to exclude in domain
The ZERO OF A FUNCTION is any replacement for the variable that will produce an answer of zero. Graphically, the real zero of a function is where the graph of the function crosses the x‐axis; that is, the real zero of a function is the x‐intercept(s) of the graph of the function Copy and paste the following link into your browser to learn more about using a graphing calculator to find the zero of a polynomial function: https://youtu.be/4aySi5mb7vc
Zero of a Polynomial Function
The domain of the inverse of f(x) = x²
[ 0 , ∞ )
(x²+8x+15)/(x+2)≥0
[-5,-3]U(-2, inf)
(2x+5) / [(x+1)(x-1)] ≥ 0
[-5/2,-1) ∪ (1,∞)
(x+6)/(x²-5x-24)≥0
[-6,-3)U(8, inf)
(x+6)/(x²+6x+8)≥0
[-6,-4)U(-2, inf)
True! these are two ways of writing the same thing.
[f o g](x) = f[g(x)] true or false?
Exponential Function
a function with a variable as the exponent
Horizontal translation
a transformation that moves a graph to the left or right
Vertical translation
a transformation that moves a graph up or down
down
a< 0, parabola opens
up
a> 0, parabola opens
f(x)=|x| 2 lines in a v shape (-1,1) (0,0) (1,1)
absolute value
Angles that have a common side and a common vertex (corner point).
adjacent angles
Domain
all possible x-values
Range
all possible y-values
Starting value
also known as initial value; represents the y-intercept
Exponential Decay
an exponential function that DECREASES from left to right
Exponential Growth
an exponential function that INCREASES from left to right
Horizontal Asymptote
an imaginary, horizontal line that a graph comes really close to, but does not cross
∆y/∆x = [ f(x₂) - f(x₁) ] / [ x₂ - x₁ ]
average rate of change
(b²-3b-28)÷(b+4)
b-7
(h,k)
circle center
(x-h)²+(y-k)²=r²
circle equation
Two angles whose sum is 90 degrees
complementary angles
root function
cube root function y=3 square root of x domain (- infinity, +positive infinity) range(- infinity, +positive infinity)
f(x)=x^3 curved line (-2,-8) (0,0) (2,8)
cubic
if we choose to remove the component (x-4) as the inner function g(x), then we replace x for every g(x) in f. h(x) = (x-4)² becomes = f[g(x)], or = f(x-4) so f(x) = x² , and g(x) = x - 4 the answer depends on which component you choose to remove.
decompose h(x) = (x-4)²
What are the domain and range of f(x)=log(x=6)-4?
domain: x > 6; range: y > -4
f(x)=e^x curved line going up with y=0 as horizontal asymptote
exponential
step1: in g(x) = √(1-x), x ≤ 1 step2: in f[g(x)] = 3/(√(1-x) - 2) step3: in 3/(√(1-x) - 2), √(1-x) ≠ 2 , so (1-x) ≠ 4, finally x ≠ -3 step4: f[g(x)], Domain {x | x ≤ 1, and x ≠ -3 }
find the domains of f[g(x)], if f(x)= 3/(x-2) and g(x)=√(1-x)
step1: in f(x) = 4x − 6, x is all real numbers step2: in g(f(x)) = √f(x) = √(4x − 6) step3: in g(f(x)) = √(4x − 6) √(4x − 6) ≥ 0, so x ≥ (3/2) step4: in g(f(x)), Domain {x | x ≥ (3/2) }
find the domains of g[f(x)], if f(x) = 4x − 6, and g(x) = √x .
y = b
form of horizontal line
x = a
form of vertical line
when you break into simpler functions from a more complicated function; finding the components of a function.
function decomposition
The inverse of f(x) = x + 11
f⁻¹(x) = x - 11
The inverse of f(x) = x³ + 1
f⁻¹(x) = ³√(x-1)
The inverse of f(x) = (x + 1)³
f⁻¹(x) = ³√x - 1
The inverse of f(x) = 2(x - 16)
f⁻¹(x) = ½x + 16
The inverse of f(x) = 2x - 16
f⁻¹(x) = ½x + 8
f(x)=[[x]] closed circle on left open on right overlapping (0,0) (1,1) (2,2) closed (1,0) (2,1) (3,2) open
greatest integer
imagine what is would look like
half circle function square root of 1-x squared domain [-1,1] range[0,1]
y=k
horizontal parabola: axis of symmetry
x=h-1/4a
horizontal parabola: directrix
(h+1/4a,k)
horizontal parabola: focus
x=a(y-k)²+h
horizontal parabola: form
|1/a| units
horizontal parabola: length of latus rectum
(h,k)
horizontal parabola: vertex
Multiply the inside of a function greater than 1
horizontal shrink
Multiply the inside of a function by a number btwn 0 & 1
horizontal stretch
Rate of change
how fast or slow a graph is changing; also known as slope for linear functions
wider
if the absolute value of a is < 1, the parabola is ____ than y=x^2
narrower
if the absolute value of a is > 1, the parabola is ___ than the graph of y=x^2
7
ln(2x-3) = ln 11
0
log (10 - 4x) = log (10 - 3x)
3
log 5x = log(2x + 9)
96
log x - log 6 = 2 log 4
4
log₂ x + log₂(x - 3) = 2
-3
log₂(x - 5) - log₂(x - 2) = 3
64
log₄ x = 3
(x₁+x₂)/2, (y₁+y₂)/2
midpoint formula
f(x)=ln(x) curved line with x=0 as vertical asymptote
natural logarithm
m
parallel slope to y = mx+b
Percent
parts per 100
- 1/m
perpendicular slope to y = mx+b
y - y₁ = m( x - x₁)
point-slope form
f(x)=x^2 parabola (-2,4) (0,0) (2,4)
quadratic
f(x)=1/x hyperbola with y=o and x=0 asymptotes (-2, -1/2) (-1,-1) (2, 1/2) (1,1)
reciprocal
m = (y₂ -y₁)/(x₂ - x₁)
slope formula
m = 0
slope of horizontal line
m is undefined
slope of vertical line
y = mx+b
slope-intercept form
f(x)=/x line going to right (0,0) (4,2) (9,3)
square root
Ax + By = C
standard / general form (linear)
Two angles whose sum is 180 degrees
supplementary angles
Base
the number or variable being raised to a power
X-intercept
the point where a graph crosses or touches the x-axis
Y-intercept
the point where a graph crosses or touches the y-axis
-b/2a
the x coordinate of a vertex standard form
opposite angles formed by two intersecting lines
vertical angles
x=h
vertical parabola: axis of symmetry
y=k-1/4a
vertical parabola: directrix
(h,k+1/4a)
vertical parabola: focus
y=a(x-h)²+k
vertical parabola: form
|1/a| units
vertical parabola: length of latus rectum
(h,k)
vertical parabola: vertex
Multiply the outside of a function by a number btwn 0 & 1
vertical shrink
Multiply the outside of a function by a number greater than 1
vertical stretch
Evaluate
when you "plug in" a number or variable
(x²+10x+24)÷(x+4)
x+6
(x²+11x+30)÷(x+5)
x+6
(x²-15x+56)÷(x-8)
x-7
(x⁴-2x²+1)÷(x²-2x+1)
x²+2x+1
Which of the following is a logarithmic function?
y=log of 3x
