Graphing Logarithmic Functions, Exponential Functions, Inverse Functions, Composite Functions, Rational and Polynomial Inequalities, Graphing Rational Functions, Rational Functions, Polynomial Division, Polynomial Functions, Quadratic Functions, Grap...

The range of the inverse of f(x) = x³

( -∞ , ∞ )

It is the set of all output values.

Range of rational functions

The domain of the inverse of f(x) = x²

[ 0 , ∞ )

(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,∞)

(7y²-3y-4)÷(y-1)

7y+4

This happens when the graph as x approaches positive infinity (+∞) or negative infinity (−∞)

End Behavior

(x+2)² < 0

No solution

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 .

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

Percent

parts per 100

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

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 is the graph of a logarithmic function?

D

The set of all real numbers except the values that are zeros of the denominator

Domain of rational functions

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 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

(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

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)

Set cancelled parts of the fraction = 0

Find holes

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?

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.

Horizontal Asymptote

an imaginary, horizontal line that a graph comes really close to, but does not cross

(b²-3b-28)÷(b+4)

b-7

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

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

(6x²+13x+6)÷(3x+2)

2x+3

(9x²-42x+45)÷(3x-8)

3x-6 - (3/3x-8)

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.

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?

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

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

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

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?

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

y= ax^2 + bx+ c

Quadratic Function Standard Form

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 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 numerator and denominator have no common factors other than positive or negative one.

Simplified Form

The degree of the numerator is more than the degree of the denominator

Slant asymptotes occur when

Domain

The set of all input values of a relation.

Range

The set of all output values of a relation.

This is an x value that a graph will approach but never touch; where the function is undefined.

Vertical Asymptote

Holes and vertical asymptotes

What to exclude in domain

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

The inverse of f(x) = 2x - 16

f⁻¹(x) = ½x + 8

Rate of change

how fast or slow a graph is changing; also known as slope for linear functions

Which of the following is a logarithmic function?

y=log of 3x