/*! 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 89 Use the functions \(f(x)=x+4\) a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 89

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$g^{-1} \circ f^{-1}$$

Short Answer

Expert verified
The composite function \(g^{-1} \circ f^{-1}(x)\) simplifies to \((x+1)/2\).

Step by step solution

01

Find the inverse function

To find the inverse of a function, first we switch places of \(x\) and \(y\). So, the inverse of function \(f(x) = y = x + 4\) will be \(x = y + 4\). Solve this for \(y\) to get \(f^{-1}(x) = x-4\). Similarly, for \(g(x) = y = 2x - 5\) is replaced with \(x = 2y - 5\). Solve for \(y\) to get \(g^{-1}(x) = (x+5) / 2\).
02

Substitute the inverse functions in the composition

Now that we have the inverse functions, we substitute them into the composition \(g^{-1} \circ f^{-1}(x)\) which becomes \(g^{-1}(f^{-1}(x)) = g^{-1}(x-4)\). This is the function \(g^{-1}(x)\) evaluated at \(x-4\). Substituting \(x-4\) for \(x\) in \(g^{-1}(x)\) we get \(g^{-1}(x-4) = \((x-4+5) /2\).
03

Simplify the expression

Simplify the expression in the numerator to get the final result. So, \(g^{-1}(x-4) = (x+1)/2\).

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

The table shows the monthly revenue \(y\) (in thousands of dollars) of a landscaping business for each month of the year \(2013,\) with \(x=1\) representing January. $$\begin{array}{|c|c|}\hline \text { Month, \(x\) } & \text { Revenue, \(y\) } \\\\\hline 1 & 5.2 \\\2 & 5.6 \\\3 & 6.6 \\ 4 & 8.3 \\\5 & 11.5 \\\6 & 15.8 \\\7 & 12.8 \\\8 & 10.1 \\\9 & 8.6 \\\10 & 6.9 \\\11 & 4.5 \\\12 & 2.7 \\\\\hline \end{array}$$ A mathematical model that represents these data is \(f(x)=\left\\{\begin{array}{l}-1.97 x+26.3 \\ 0.505 x^{2}-1.47 x+6.3\end{array}\right.\) (a) Use a graphing utility to graph the model. What is the domain of each part of the piecewise-defined function? How can you tell? Explain your reasoning. (b) Find \(f(5)\) and \(f(11),\) and interpret your results in the context of the problem. (c) How do the values obtained from the model in part (a) compare with the actual data values?

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$g(x)=|x|-5$$

Finding a Mathematical Model In Exercises \(41-50\), find a mathematical model for the verbal statement. Newton's Law of Cooling: The rate of change \(R\) of the temperature of an object is directly proportional to the difference between the temperature \(T\) of the object and the temperature \(T_{e}\) of the environment in which the object is placed.

Determine whether the statement is true or false. Justify your answer. Every function is a relation.

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 220 newtons stretches a spring 0.12 meter. What force is required to stretch the spring 0.16 meter?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

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.