/*! 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 73 Describe how to use the graph of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 73

Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.

Short Answer

Expert verified
To graph the inverse of a one-to-one function, reflect the graph of the original function over the line \(y = x\). This means for every point (x, y) on the graph of the original function, there will be a point (y, x) on the graph of the inverse function. Also transpose any intercepts and asymptotes.

Step by step solution

01

Understanding One-to-One Functions

A function is said to be one-to-one if every x value corresponds to exactly one y value, and every y value corresponds to exactly one x value. This means no two different x-values in the domain map to the same y-value in the range and vice versa.
02

Understanding Inverse Functions

An inverse function is a function which undoes the operation of the original function. In simpler terms, if you have a function that takes x to y, then its inverse function takes y back to x. If \(f(x)\) is the original function then the inverse function is usually written as \(f^{-1}(x)\). For a function to have an inverse, it must be a one-to-one function.
03

Graphing the Inverse

To graph the inverse of a function, reflect the graph of the original function over the line \(y = x\). This is because in the inverse function, the roles of x and y are interchanged, hence the reflection over the line \(y = x\). For every point (a,b) on the graph of \(f(x)\), there will be a point (b,a) on the graph of \(f^{-1}(x)\)
04

Key points

Key points to note in the graphs of function and its inverse are the intercepts and any asymptotes. The x-intercept of the function becomes y-intercept of the inverse function and vice versa. Any horizontal asymptote on the function's graph becomes a vertical asymptote on the inverse's graph and vice versa.

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

A department store has two locations in a city. From 2012 through \(2016,\) the profits for each of the store's two branches are modeled by the functions \(f(x)=-0.44 x+13.62\) and \(g(x)=0.51 x+11.14 .\) In each model, \(x\) represents the number of years after \(2012,\) and \(f\) and \(g\) represent the profit, in millions of dollars. a. What is the slope of \(f ?\) Describe what this means. b. What is the slope of \(g\) ? Describe what this means. c. Find \(f+g .\) What is the slope of this function? What does this mean?

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{\frac{1}{3}}(x-4)$$

?. The regular price of a pair of jeans is \(x\) dollars. Let \(f(x)=x-5\) and \(g(x)=0.6 x\) a. Describe what functions \(f\) and \(g\) model in terms of the price of the jeans. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the jeans. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the jeans, \(f \circ g\) or \(g \circ f ?\) Explain.

A telephone company offers the following plans. Also given are the piecewise functions that model these plans. Use this information to solve. Plan \(A\) \(\cdot \$ 30\) per month buys 120 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$C(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 120 \\\30+0.30(t-120) & \text { if } t>120 \end{array}\right. $$ Plan \(B\) \(\cdot \ 40\) per month buys 200 minutes. \(\cdot\) Additional time costs \(\$ 0.30\) per minute. $$ C(t)=\left\\{\begin{array}{ll} 40 & \text { if } 0 \leq t \leq 200 \\\ 40+0.30(t-200) & \text { if } t>200 \end{array}\right. $$ Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$(y+1)^{2}=36-(x-3)^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.