/*! 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 26 Use the Round-Trip Theorem on pa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 26

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=\frac{-3}{2 x+5}, \quad g(x)=\frac{-3-5 x}{2 x}$$

Short Answer

Expert verified
Question: Show that g is the inverse of f using the Round-Trip Theorem, where f(x) = (-3) / (2x+5) and g(x) = (-3-5x) / (2x). Answer: Since we have proven that both f(g(x)) = x and g(f(x)) = x, it confirms that g is indeed the inverse of f, according to the Round-Trip Theorem.

Step by step solution

01

Calculate f(g(x))

To find f(g(x)), we need to substitute g(x) into the expression for f(x): $$ f(g(x)) = f\left(\frac{-3-5x}{2x}\right) = \frac{-3}{2 \left(\frac{-3-5x}{2x}\right) +5} $$ Now, let's simplify this expression.
02

Simplify f(g(x))

We need to simplify the denominator by adding the term inside the parenthesis: $$ \begin{aligned} f(g(x)) &= \frac{-3}{2 \left(\frac{-3-5x}{2x}\right) +5}\\ &= \frac{-3}{\frac{-6-10x}{2x}+5} \end{aligned} $$ Now, let's find a common denominator and combine the terms in the denominator: $$ \begin{aligned} f(g(x)) &= \frac{-3}{\frac{-6-10x+10x}{2x}}\\ &= \frac{-3}{\frac{-6}{2x}} \end{aligned} $$ Finally, let's simplify the fraction: $$ \begin{aligned} f(g(x)) &= \frac{-3}{\frac{-6}{2x}}\\ &= \frac{-3}{-3} \cdot \frac{2x}{1}\\ &= x \end{aligned} $$ Now that we have found f(g(x)) = x, we need to find g(f(x)).
03

Calculate g(f(x))

To find g(f(x)), we need to substitute f(x) into the expression for g(x): $$ g(f(x)) = g\left(\frac{-3}{2x+5}\right) = \frac{-3-5\left(\frac{-3}{2x+5}\right)}{2\left(\frac{-3}{2x+5}\right)} $$ Now, let's simplify this expression.
04

Simplify g(f(x))

We need to simplify the numerator and the denominator: $$ \begin{aligned} g(f(x)) &= \frac{-3-5\left(\frac{-3}{2x+5}\right)}{2\left(\frac{-3}{2x+5}\right)}\\ &= \frac{-3+\frac{15}{2x+5}}{\frac{-6}{2x+5}} \end{aligned} $$ Now, let's find a common denominator and combine the terms in the numerator: $$ \begin{aligned} g(f(x)) &= \frac{\frac{-3(2x+5)+15}{2x+5}}{\frac{-6}{2x+5}}\\ &= \frac{-6x-15}{2x+5} \cdot \frac{2x+5}{-6}\\ \end{aligned} $$ Finally, let's simplify the fraction: $$ \begin{aligned} g(f(x)) &= \frac{-6x-15}{2x+5} \cdot \frac{2x+5}{-6}\\ &= \frac{-1}{-1}\cdot \frac{6x+15}{6}\\ &= x \end{aligned} $$
05

Conclusion

Since we have shown that f(g(x)) = x and g(f(x)) = x, we can conclude that g is the inverse of f using the Round-Trip Theorem.

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.

Round-Trip Theorem
The Round-Trip Theorem is a fundamental principle in understanding inverse functions. This theorem states that if you have two functions, say f and g, and g is the inverse of f, then when you compose f with g (and vice versa), the result should bring you back to where you started, essentially making a 'round trip'. In mathematical terms, this means that f(g(x)) = x and g(f(x)) = x.

Applying this to our example, we calculated f(g(x)) and found it simplifies to x, and then calculated g(f(x)) and also found it simplifies to x. This two-way successful round trip confirms that g and f are indeed inverses of each other according to the Round-Trip Theorem.
Function Composition
Function composition involves combining two functions in a way that the output of one function becomes the input of the other. In the notation f(g(x)), the function g(x) is applied first, and its output is then used as the input for the function f. This creates a new function that can then be simplified or analyzed for various properties.

In the exercise, we first composed f with g by substituting g(x) into f(x), and then we composed g with f in the same way. Through these compositions, we were aiming to demonstrate that each function 'undoes' the effect of the other, which is a key characteristic of inverse functions.
Simplifying Expressions
Simplifying expressions is the process of reducing a complex mathematical expression into a simpler or more comprehensible form without changing its value. The goal is to make the expression easier to understand or work with. This involves combining like terms, finding common denominators, and canceling terms when possible.

In the provided solution, simplification plays an essential role. For both f(g(x)) and g(f(x)), simplifying the complex fractions was required in order to reveal their simplest form, x. Without simplification, it wouldn鈥檛 have been as apparent that g is indeed the inverse of f. It's like untangling a knot to reveal a straight line.
Verification of Inverse Functions
Verification of inverse functions is the process of proving that a given function g is the inverse of another function f. This can be accomplished through several methods, but one of the most straightforward is the use of the Round-Trip Theorem.

In our exercise, we've employed this theorem for verification by showing f(g(x)) = x and g(f(x)) = x. By doing this, we have effectively verified that f and g are inverses. It's a powerful confirmation because it doesn't just suggest but rather conclusively proves that g undoes whatever f does, and vice versa. This kind of verification is critical in precalculus because it ensures the functions have the proper relationship to be considered inverses.

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

Write the rule of a function g whose graph can be obtained from the graph of the function \(f\) by performing the transformations in the order given. \(f(x)=x^{2}+2 ;\) shift the graph horizontally 5 units to the left and then vertically upward 4 units.

Find the indicated values of the function by hand and by using the table feature of a calculator (or the EVAL key on TI-85/86). If your answers do not agree with each other or with those at the back of the book, you are either making algebraic mistakes or incorrectly entering the function in the equation memory. \(f(x)=\frac{x-3}{x^{2}+4}\) (d) \(f(2)\) (a) \(f(-1)\) (b) \(f(0)\) (c) \(f(1)\) (e) \(f(3)\)

Find the radius \(r\) and height \(h\) of a cylindrical can with a surface area of 60 square inches and the largest possible volume, as follows. (a) Write an equation for the volume \(V\) of the can in terms of \(r\) and \(h\). (b) Write an equation in \(r\) and \(h\) that expresses the fact that the surface area of the can is \(60 .\) [ Hint: Think of cutting the top and bottom off the can; then cut the side of the can lengthwise and roll it out flat; it's now a rectangle. The surface area is the area of the top and bottom plus the area of this rectangle. The length of the rectangle is the same as the circumference of the original can (why?).] (c) Write an equation that expresses \(V\) as a function of \(r\) [Hint: Solve the equation in part (b) for \(h\), and substitute the result in the equation of part (a).] (d) Graph the function in part (c), and find the value of \(r\) that produces the largest possible value of \(V\). What is \(h\) in this case?

The integer part function has the set of all real numbers (written as decimals) as its domain. The rule is "For each input number, the output is the part of the number to the left of the decimal point." For instance, the input 37.986 produces the output \(37,\) and the input -1.5 produces the output \(-1 .\) On most calculators, the integer part function is denoted "iPart." On calculators that use "Intg" or "Floor" for the greatest integer function, the integer part function is denoted by "INT." (a) For each nonnegative real number input, explain why both the integer part function and the greatest integer function [Example \(7]\) produce the same output. (b) For which negative numbers do the two functions produce the same output? (c) For which negative numbers do the two functions produce different outputs?

Jack and Jill are salespersons in the suit department of a clothing store. Jack is paid \(\$ 200\) per week plus \(\$ 5\) for each suit he sells, whereas Jill is paid \(\$ 10\) for every suit she sells. (a) Let \(f(x)\) denote Jack's weekly income, and let \(g(x)\) denote Jill's weekly income from selling \(x\) suits. Find the rules of the functions \(f\) and \(g\). (b) Use algebra or a table to find \(f(20)\) and \(g(20), f(35)\) and \(g(35), f(50)\) and \(g(50)\) (c) If Jack sells 50 suits a week, how many must Jill sell to have the same income as Jack?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Probability and 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.