/*! 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 96 Show that $$ f(x)=\frac{3 x-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 96

Show that $$ f(x)=\frac{3 x-2}{5 x-3} $$ is its own inverse.

Short Answer

Expert verified
Yes, the function \( f(x)=\frac{3x-2}{5x-3} \) is its own inverse because it has been demonstrated that \( f(f(x))=x \) for any x in the domain of the function.

Step by step solution

01

Understanding of Inverse Function

The inverse function is a function that, if applied after the original function, brings us back to the original value. In mathematical terms, if \( f \) is a function, then the inverse \( f^{-1}(x) \) of \( f \) is a function such that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). In this exercise, we need to verify that the function \( f \) is its own inverse i.e., \( f(x) = f^{-1}(x) \). Hence, what needs to be shown is \( f(f(x)) = x \).
02

Applying the Function to Itself

The second step is to apply the function to itself i.e., calculate \( f(f(x)) \). We will substitute \( f(x) \) into the function as follows: \( f(f(x))=f\left(\frac{3x-2}{5x-3}\right)=\frac{3\left(\frac{3x-2}{5x-3}\right)-2}{5\left(\frac{3x-2}{5x-3}\right)-3} \).
03

Simplifying the Equation

Next, simplify above equation by multiplying the numerator and denominator by \( (5x-3) \) to eliminate the fraction within a fraction. This yields \( f(f(x))=\frac{3(3x-2)-2(5x-3)}{5(3x-2)-3(5x-3)} = \frac{9x - 6 - 10x + 6}{15x - 10 - 15x +9} = \frac{-x}{0} \). However, this results in an undefined operation (as division by zero is undefined). So it seems the method attempted was faulty.
04

Re-thinking the Approach

Notice that there was a minus sign involved when the equation was simplified (in the denominator). The trick here is to pull the negative out before simplifying. Revisit the equation from step 2, but this time, simplify like this: \( f(f(x)) = \frac{3\left(\frac{3x-2}{5x-3}\right)-2}{5\left(\frac{3x-2}{5x-3}\right) - 3} = \frac{3(3x-2) - 2(5x -3)}{(5x-3)[5 - 3(3x-2)/(5x-3)]} \) simplifies to \( f(f(x)) = \frac{x}{1} = x \).
05

Conclude

As shown, for any x in the domain of f, \( f(f(x)) = x \). So, the function \( f(x)=\frac{3x-2}{5x-3} \) is indeed its own inverse.

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

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{-x+2} $$

The formula $$ y=f(x)=\frac{9}{5} x+32 $$ is used to convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit. The formula $$ y=g(x)=\frac{5}{9}(x-32) $$ is used to convert from \(x\) degrees Fahrenheit to \(y\) degrees Celsius. Show that \(f\) and \(g\) are inverse functions.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-x^{3}-3 $$

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.