1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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Finding the Inverse of a Function or Showing One Does Not Exist, Ex 1

Graph the inverse of each one-to-one function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/e969a89d7eb2fda8-1679520514618.jpg" alt="" />

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = = -x and g(x) = -x

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = x³ +2

Inverse Functions - The Basics!

Determine whether each function graphed or defined is one-to-one. y = 2x^3 - 1

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = x³ − 1

Find the inverse of each function that is one-to-one. {(3,-1), (5,0), (0,5), (4, 2/3)}

Determine whether each function graphed or defined is one-to-one. y = 2x - 8

In Exercises 59-64, let
f(x) = 2x - 5 
g(x) = 4x - 1
h(x) = x² + x + 2.
Evaluate the indicated function without finding an equation for the function. ƒ¹ (1)

Determine whether each pair of functions graphed are inverses. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/35bc5eea803ae15f-1680899598830.jpg" alt="" />

Determine whether each function graphed or defined is one-to-one. y = ∛x+1 - 3

Graph the inverse of each one-to-one function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/4fdf5e850bb6e774-1679517426503.jpg" alt="" />

Graphing and determining the inverse of a function

Which graphs in Exercises 29–34 represent functions that have inverse functions?

Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = x^2+3, x≥0; g(x) = √x-3, x≥3

Determine whether each function graphed or defined is one-to-one. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/7a4f6d22f6c80441-1680894943870.jpg" alt="" />

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 3/(x-4) and g(x) = 3/x + 4

The Horizontal Line Test

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range of f and ƒ¯¹. f(x)=2x-1

Determine whether each pair of functions graphed are inverses. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/d1665d327c8a03f4-1681133194800.jpg" alt="" />

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x − 1)², x ≤ 1

Determine whether each function graphed or defined is one-to-one. y = x+4 / x-3

Verifying Functions are Inverses

In Exercises 101–102, find an equation for f^(-1)(x). Then graph f and f^(-1) in the same rectangular coordinate system. f(x) = 1 - x^2, x ≥ 0.

Finding the Inverse of a Function or Showing One Does not Exist, Ex 4

Find the Inverse of a Function and Graph Both Functions

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=5x-9 and g(x) = (x+5)/9

The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^(-1)x, the inverse function. (b) Verify that your equation is correct by showing that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. f(x) = (x - 7)/(x + 2)

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = √(x-1)

Graph the inverse of each one-to-one function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/fb6f371e84f4cda1-1679488549878.jpg" alt="" />

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range of f and ƒ¯¹. ƒ(x) = x² − 4, x ≥ 0

Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = 2/x+6, g(x) = 6x+2/x

Horizontal line test

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 4x + 9 and g(x) = (x-9)/4

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = 2x + 3

Find the inverse of f(x)=(x−10)/(x+10).

Finding an Inverse by Interchange of Variables

Determine whether each pair of functions graphed are inverses. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/f359ec5e9ee220d5-1680900210428.jpg" alt="" />

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x +4)/(x-2)

Graphs of Inverses

Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = -4x+2, g(x) = -1/4x - 2

Determine whether each function graphed or defined is one-to-one. y = -1 / x+2

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = ∛(x − 4) and g(x) = x³ +4

Which graphs in Exercises 96–99 represent functions that have inverse functions?

Determine whether each function graphed or defined is one-to-one. y = 2(x+1)^2 - 6

Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = x+1/x-2, g(x) = 2x+1/x-1

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x+2)³

Determine whether each function graphed or defined is one-to-one. y = 5|x+2|

Determine whether each function graphed or defined is one-to-one. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/444c200b77974e8b-1680895049442.jpg" alt="" />

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = √x

Exercises 123–125 will help you prepare for the material covered in the next section. Solve for y: x = y² -1, y ≥ 0.

Finding an Inverse Function by Undoing Steps

The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^(-1)x, the inverse function. (b) Verify that your equation is correct by showing that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. f(x) = 4x - 3

Finding the Inverse of a Function or Showing One Does not Exist, Ex 3

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = x +3

Exercises 123–125 will help you prepare for the material covered in the next section. Solve for y : x = 5/y + 4

Graphing the inverse of a function given its graph

Graph the inverse of each one-to-one function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/27b454dd3c89305c-1679428995873.jpg" alt="" />

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = 2x

Graph the inverse of each one-to-one function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/3e33e866fba48f4b-1679431136818.jpg" alt="" />

Horizontal Line Test and One to One Functions

Determine whether the given functions are inverses. ƒ= {(2,5), (3,5), (4,5)}; g = {(5,2)}

If f(x) = 3x and g(x) = x + 5, find (ƒ 0 g)¯¹ (x) and (g¯¹ o ƒ˜¹) (x).

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = 1/x

Graph the inverse of each one-to-one function. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/1b5444e857148494-1679427689532.jpg" alt="" />

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3

Verify Inverse Functions

Inverse Functions

Verifying inverse functions by composition: not inverse | High School Math | Khan Academy

Determine whether each pair of functions graphed are inverses. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/2fd994fb35dcd04f-1679423213715.jpg" alt="" />

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = ∛x + 1

Find the inverse of each function that is one-to-one. {(1, -3), (2, -7), (4, -3), (5, -5)}

Determine whether each function graphed or defined is one-to-one. y = -√100 - x^2

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (2x +1)/(x-3)

Determine whether the given functions are inverses.

Determine whether each function graphed or defined is one-to-one. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/f213e346558c459c-1680895185573.jpg" alt="" />

In Exercises 39-52,
a. Find an equation for ƒ¯¹(x).
b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system.
c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x+2)³

In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 4x and g(x) = x/4

Finding the Inverse of a Function or Showing One Does not Exist, Ex 2

Determine whether each function graphed or defined is one-to-one. y = -√x+5

Use the Graph of a One-to-One Function to Graph its Inverse Function on the Same Axes

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