Skip to main content
Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 51abc
Chapter 3, Problem 51abc

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

Verified step by step guidance
1
Step 1: To find the inverse function ƒ¯¹(x), start by replacing f(x) with y. This gives y = ∛x + 1. Next, swap x and y to reflect the inverse relationship. This results in x = ∛y + 1.
Step 2: Solve for y in terms of x. Subtract 1 from both sides to isolate the cube root term: x - 1 = ∛y. Then, cube both sides to eliminate the cube root: (x - 1)^3 = y. Thus, the inverse function is ƒ¯¹(x) = (x - 1)^3.
Step 3: To graph f(x) = ∛x + 1 and its inverse ƒ¯¹(x) = (x - 1)^3 on the same coordinate system, note that the graph of an inverse function is a reflection of the original function across the line y = x. Plot several points for both functions and ensure symmetry about the line y = x.
Step 4: Determine the domain and range of f(x). The cube root function ∛x is defined for all real numbers, and adding 1 does not restrict the domain. Therefore, the domain of f(x) is (-∞, ∞). The range of f(x) is also (-∞, ∞) because the cube root function can output all real numbers.
Step 5: Determine the domain and range of ƒ¯¹(x). Since the inverse function reflects the domain and range of the original function, the domain of ƒ¯¹(x) is (-∞, ∞), and the range of ƒ¯¹(x) is also (-∞, ∞). Use interval notation to express these results.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
10m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f¯¹(y) takes y back to x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y. Understanding this concept is crucial for finding f¯¹(x) in the given problem.
Recommended video:
4:30
Graphing Logarithmic Functions

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the relationship between the input (x) and output (f(x)). For the original function f(x) and its inverse f¯¹(x), their graphs will reflect across the line y = x. This symmetry is key to understanding how the two functions relate to each other and aids in accurately graphing both functions.
Recommended video:
5:26
Graphs of Logarithmic Functions

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (f(x)). For the function f(x) = ∛x + 1, the domain is all real numbers since cube roots are defined for all x, and the range is also all real numbers due to the nature of the cube root function. Understanding domain and range is essential for determining the characteristics of both f and its inverse.
Recommended video:
4:22
Domain & Range of Transformed Functions
Related Practice
Textbook Question

Use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)

Textbook Question

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Textbook Question

In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. f(g(1))

Textbook Question

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g(x) = √x −1 (x = 0, 1, 4, 9)

Textbook Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 1)² + y² = 25

2
views
Textbook Question

Find

a. (fog) (x)

b. (go f) (x)

c. (fog) (2)

d. (go f) (2).

f(x) = 2x, g(x) = x+7