/*! 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 53 In Exercises \(49-62,\) (a) find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 53

In Exercises \(49-62,\) (a) find the inverse function of \(f\) (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\) , and (d) state the domain and range of \(f\) and \(f^{-1}\) . $$ f(x)=\sqrt{4-x^{2}}, \quad 0 \leq x \leq 2 $$

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \sqrt{4 - x^2}\). Both \(f(x)\) and \(f^{-1}(x)\) are plotted as semi-circles. They reflect about the line \(y = x\). The domain and range for both the original function and its inverse are \(0 \leq x \leq 2\) and \(0 \leq y \leq 2\), respectively.

Step by step solution

01

Find the Inverse Function

We start by setting \(y = f(x)\), yielding the equation \(y = \sqrt{4-x^2}\). To find the inverse, we switch \(y\) and \(x\), then solve it for \(y\). This gives us \(x = \sqrt{4 - y^2}\). Solving for \(y\) gives us \(y = \sqrt{4 - x^2}\) and \(y = -\sqrt{4 - x^2}\). Since the given domain is \(0 \leq x \leq 2\), the inverse function becomes \(f^{-1}(x) = \sqrt{4 - x^2}\).
02

Graph

Next, we graph both functions. \(f(x)\) is a semi-circle above the \(x\)-axis with radius 2, while \(f^{-1}(x)\) is a semi-circle to the right of the \(y\)-axis with the same radius. These graphs are mirror images over the line \(y=x\).
03

Describe the Relationship

The function \(f(x)\) and its inverse \(f^{-1}(x)\) are reflections of each other across the line \(y = x\). When a function has an inverse, this is always the relationship between their graphs.
04

Determine the Domain and Range

The original function \(f(x)\) is defined for \(0 \leq x \leq 2\), with range \(0 \leq y \leq 2\). For the inverse function \(f^{-1}(x)\), the domain and range reverse, making the domain \(0 \leq x \leq 2\) and the range \(0 \leq y \leq 2\).

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!

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

SPORTS The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. \((1948, 5.30)\) \((1952, 5.20)\) \((1956, 4.91)\) \((1960, 4.84)\) \((1964, 4.72)\) \((1968, 4.53)\) \((1972, 4.32)\) \((1976, 4.16)\) \((1980, 4.15)\) \((1984, 4.12)\) \((1988, 4.06)\) \((1992, 4.12)\) \((1996, 4.12)\) \((2000, 4.10)\) \((2004, 4.09)\) \((2008, 4.05)\) A linear model that approximates the data is \(y = -0.020t + 5.00\), where \(y\) represents the winning time (in minutes) and \(t=0\) represents 1950. Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)

THINK ABOUT IT In Exercises 77-86, restrict the domain of the function \(f\) so that the function is one-to-one and has an inverse function. Then find the inverse function \(f^{-1}\). State the domains and ranges of \(f\) and \(f^{-1}\). Explain your results. (There are many correct answers.) \(f(x) = |x-4| + 1\)

THINK ABOUT IT The function given by \(f(x) = k(2 - x - x^3)\) has an inverse function, and \(f^{-1}(3) = -2\). Find \(k\).

Mathematical models that involve both direct and inverse variation are said to have ________ variation.

THINK ABOUT IT In Exercises 77-86, restrict the domain of the function \(f\) so that the function is one-to-one and has an inverse function. Then find the inverse function \(f^{-1}\). State the domains and ranges of \(f\) and \(f^{-1}\). Explain your results. (There are many correct answers.) \(f(x) = \frac{1}{2}x^2 - 1\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.