/*! 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 40 For the following exercises, fin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 40

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=\frac{1}{x^{2}}, x \geq 0 $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), for \( x > 0 \).

Step by step solution

01

Understanding the Function

We have the function \( f(x) = \frac{1}{x^2} \) with the condition \( x \geq 0 \). This means the function is defined for non-negative values of \( x \).
02

Setting Up for Inverse Function

To find the inverse, we start by replacing \( f(x) \, \text{with} \, y \). This gives us \( y = \frac{1}{x^2} \). Our job is to solve for \( x \) in terms of \( y \).
03

Solving for x in Terms of y

Rearrange the equation to express \( x \) in terms of \( y \): \( y = \frac{1}{x^2} \) can be rewritten as \( x^2 = \frac{1}{y} \). Solving for \( x \), we get \( x = \sqrt{\frac{1}{y}} \), and since \( x \geq 0 \), \( x = \frac{1}{\sqrt{y}} \). Thus, the inverse function is \( f^{-1}(y) = \frac{1}{\sqrt{y}} \) for \( y > 0 \).
04

State the Inverse Function

The inverse function is \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), defined for \( x > 0 \).
05

Graphing the Function and Its Inverse

Graph the original function \( y = \frac{1}{x^2} \) and its inverse \( y = \frac{1}{\sqrt{x}} \). The graphs will reflect across the line \( y = x \).

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.

Function Transformation
Function transformation involves modifying a function to achieve a desired form. For the function given in the exercise, which is \( f(x) = \frac{1}{x^2} \) for \( x \geq 0 \), transformation plays a key role when finding the inverse function. When working with transformations, keep these points in mind:
  • The inverse transformation exchanges the roles of \( x \) and \( y \), essentially flipping the function across the line \( y = x \).
  • This specific transformation involves algebraic manipulation, which directly leads to finding the inverse.
  • Apply transformations carefully to correctly hold onto any restrictions, such as \( x \geq 0 \) in this exercise.
The process of switching \( x \) and \( y \) reflects the core principle of finding inverse functions, directly impacting the transformation applied to the function.
Graphing Functions
Graphing functions makes abstract mathematical concepts more tangible. When graphing \( f(x) = \frac{1}{x^2} \), it is important to remember:
  • The function is only defined for \( x \geq 0 \).
  • As \( x \) increases, \( f(x) \) decreases, approaching zero, but never reaching it.
  • The graph is a downward opening parabola that is steepest near the origin on the positive side.
Graphing its inverse \( f^{-1}(x) = \frac{1}{\sqrt{x}} \) involves similar steps and helps visualize the relationship between a function and its inverse.
To graph the inverse:
  • The inverse is defined for \( x > 0 \) and decreases as \( x \) increases.
  • Plotting both graphs on the same axes illustrates their reflection symmetry across the line \( y = x \).
  • Symmetry confirms the correctness of inverse calculations and provides visual insight into their relationships.
Domain and Range
Understanding the domain and range of a function is essential for graphing both the function and its inverse. For the function \( f(x) = \frac{1}{x^2} \):
  • Domain: Since \( x \geq 0 \), the domain is \( [0, \infty) \).
  • Range: As \( f(x) \) \( \geq 0 \) for all \( x \), the range is \((0, \infty) \).
For the inverse function \( f^{-1}(x) = \frac{1}{\sqrt{x}} \):
  • Domain: Since \( x > 0 \), the domain is \( (0, \infty) \).
  • Range: The range aligns with the domain of the original function: \( [0, \infty) \).
Understanding these relationships between the original function and its inverse ensures appropriately defined graphs on a coordinate plane.
Calculus
Although the task at hand primarily involves algebraic transformations and graphing, calculus often plays a supporting role. In this context, calculus helps us:
  • Analyze the behavior of both the function and its inverse, particularly their limits as they approach endpoints of their domains.
  • Confirm that the transformations applied hold true across all required intervals.
  • Calculate derivative functions, which can further deepen understanding of the function's behavior, although not explicitly covered here.
Using calculus can improve intuition and verification when working through the relationships and transformations necessary with inverse functions.

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, find the inverse of the functions with \(a,\) c positive real numbers. $$f(x)=\sqrt[3]{a x+b}$$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the square root of \(z\). When \(x=2\) and \(z=9\), then \(y=24\). Find \(y\) when \(x=3\) and \(z=25\).

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the cube of \(z\) and inversely as the square root of \(w\). When \(x=2\), \(z=2\), and \(w=64\), then \(y=12\). Find \(y\) when \(x=1\), \(z=3,\) and \(w=4\).

For the following exercises, list all possible rational zeros for the functions. \(f(x)=6 x^{4}-10 x^{2}+13 x+1\)

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube root of \(x\). When \(x=27,\) then \(y=5 .\) Find \(y\) when \(x=125 .\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.