/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 190 For the following exercises, a. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 190

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. $$ f(x)=\sqrt[3]{x-4} $$

Short Answer

Expert verified
The inverse is \( f^{-1}(x) = x^3 + 4 \). The domain and range are both \(-\infty, \infty\).

Step by step solution

01

Set up the equation for the inverse function

To find the inverse function, we start by considering the equation given: \[ f(x) = \sqrt[3]{x-4} \]We need to express \( f(x) \) in terms of \( y \):\[ y = \sqrt[3]{x-4} \]
02

Solve for x in terms of y

We will manipulate the equation to express \( x \) in terms of \( y \). Start by eliminating the cube root on the right side of the equation:\[ y^3 = x - 4 \]Next, solve for \( x \) by adding 4 to both sides:\[ x = y^3 + 4 \]
03

Express the inverse function

Now that we have \( x \) in terms of \( y \), interchange \( x \) and \( y \) to express the inverse function. Replace \( y \) with \( f^{-1}(x) \) and \( x \) with \( y \) in the equation:\[ f^{-1}(x) = x^3 + 4 \]
04

Determine the domain of the inverse function

The domain of the inverse function \( f^{-1}(x) = x^3 + 4 \) is all real numbers because any real number \( x \) can be cubed and then added to 4. Therefore, the domain is:\(-\infty, \infty\)
05

Determine the range of the inverse function

The range of the inverse function is also all real numbers. Since a cubic function can take any real number as input and produce any real number as output, the range is:\(-\infty, \infty\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cubic Function
Cubic functions are a special type of polynomial function where the highest degree is three. This means the function will have the form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). In the context of our exercise, when finding the inverse, we specifically dealt with the function \( f(x) = \sqrt[3]{x-4} \). After manipulation, this function is transformed to \( f^{-1}(x) = x^3 + 4 \).
Key characteristics of cubic functions include:
  • They have one inflection point, where the curve changes concavity.
  • The graph can cross the x-axis up to three times, depending on its roots.
  • Cubic functions extend infinitely in both vertical directions as \( x \) becomes infinitely large or small.
Understanding these properties helps in visualizing how the graph might look, and why the range of such functions often covers all real numbers.
Domain and Range
When analyzing functions, the domain represents all the possible inputs (\( x \)-values), while the range represents all possible outputs (\( y \)-values). In this exercise, we worked with inverse functions. After finding the inverse function \( f^{-1}(x) = x^3 + 4 \), it is crucial to determine its domain and range.
  • The domain of \( f^{-1}(x) \) is all real numbers. Since you can potentially cube any real number and add four, there are no restrictions on \( x \).
  • The range of \( f^{-1}(x) \) is also all real numbers. A cubic function can indeed reach any positive or negative value as output, reflecting symmetry and behavior on the graph.
This comprehensive domain and range suggest that there are no limits beyond the real-number set, a property typically seen in polynomial functions of an odd degree.
Function Manipulation
Function manipulation is a key skill in solving problems involving inverse functions. It involves changing the function鈥檚 equation to uncover new relationships or solve for variables.
In the exercise, we transformed \( f(x) = \sqrt[3]{x-4} \) into its inverse \( f^{-1}(x) = x^3 + 4 \). This required:
  • Eliminating operations: Start by removing the original cube root by cubing both sides of the equation, isolating the term inside the root.
  • Solving for the variable: Rearrange terms to solve for the variable originally inside the root, ensuring that you're expressing it in terms of the new variable.
  • Interchanging variables: Finally, swap the roles of x and y to derive the inverse, aligning with the functions鈥� symbolic expressions.
This kind of manipulation not only helps find inverses but also deepens understanding of function transformations, making problem-solving with functions more intuitive.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the following exercises, for each of the piecewise- defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph. $$ g(x)=\left\\{\begin{array}{l}{\frac{3}{x-2}, x \neq 2} \\ {4, x=2}\end{array} ; g(0) ; g(-4) ; g(2)\right. $$

The amount \(A\) of a \(\$ 100,000\) investment paying continuously and compounded for \(t\) years is given by \(A(t)=100,000 \cdot e^{0.055 t} .\) Find the amount \(A\) accumulated in 5 years.

Simplify the following trigonometric expressions. $$ \frac{\tan ^{2} x}{\sec ^{2} x}+\cos ^{2} x $$

For the following exercises, solve the exponential equation exactly. $$ e^{3 x}-15=0 $$

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=1-\ln x $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.