/*! 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 For each function as defined tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 73

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form y = 茠 -11x2, (b) graph 茠 and 茠 -1 on the same axes, and (c) give the domain and the range of 茠 and 茠 -1. If the function is not one-to-one, say so. $$f(x)=\frac{-3 x+12}{x-6}, \quad x \neq 6$$

Short Answer

Expert verified
(a) \( f^{-1}(x) = \frac{12+6x}{x+3} \), (b) Graph the function and its inverse symmetrically along \( y=x \), (c) Domain of \( f \) is \( x eq 6 \), range is \( y eq -3 \); Domain of \( f^{-1} \) is \( x eq -3 \), range is \( y eq 6 \).

Step by step solution

01

Verify One-to-One Function

Check if the function \( f(x) = \frac{-3x+12}{x-6} \) is one-to-one by ensuring it passes the Horizontal Line Test.
02

Write the Equation for the Inverse Function

Start by setting \( y = \frac{-3x+12}{x-6} \). Then solve for \( x \) to express it in terms of \( y \). Swap \( x \) and \( y \) at the end to find the inverse.
03

Algebraic Manipulation

Multiply both sides by \( x-6 \): \[ y(x-6) = -3x + 12 \]Distribute and rearrange terms to isolate \( x \): \[ yx - 6y = -3x + 12 \]Move all \( x \)-terms to one side: \[ yx + 3x = 12 + 6y \]Factor out \( x \): \[ x(y + 3) = 12 + 6y \]Solve for \( x \): \[ x = \frac{12 + 6y}{y + 3} \]Thus, the inverse function is: \[ f^{-1}(x) = \frac{12 + 6x}{x + 3} \]
04

Graph the Function and its Inverse

Graph \( f(x) = \frac{-3x+12}{x-6} \) and \( f^{-1}(x) = \frac{12+6x}{x+3} \) on the same axes. The graphs should be symmetrical along the line \( y = x \).
05

Determine the Domain and Range

For \( f(x) = \frac{-3x+12}{x-6} \), the domain is \( x eq 6 \) and the range is \( y eq -3 \). For \( f^{-1}(x) = \frac{12+6x}{x+3} \), the domain is \( x eq -3 \) and the range is \( y eq 6 \).

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.

One-to-One Functions
A function is one-to-one if every output value corresponds to exactly one input value. This means different inputs produce different outputs. Checking if a function is one-to-one is crucial before finding its inverse.

To determine if a function is one-to-one, you can use the horizontal line test.

If no horizontal line intersects the graph of the function more than once, the function is one-to-one.

For the function \( f(x) = \frac{-3x+12}{x-6} \), we need to check if it passes the horizontal line test.
Horizontal Line Test
The horizontal line test is a visual way to determine if a function is one-to-one. Imagine drawing horizontal lines all over the graph of the function:

- If any horizontal line crosses the function more than once, it's not a one-to-one function.
- If no horizontal line crosses more than once, the function is one-to-one.

This test works because it ensures each output is associated with a single input, which is the essence of a one-to-one function.

For our function \( f(x) = \frac{-3x+12}{x-6} \), we can graph it to verify:

- The function \( f(x) = \frac{-3x+12}{x-6} \) passes the horizontal line test, so it is one-to-one.

Now we can find its inverse.
Domain and Range
When discussing functions, two important concepts are domain and range.

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

For the function \( f(x) = \frac{-3x+12}{x-6} \):

  • The domain excludes the value that makes the denominator zero, so the domain is \( x eq 6 \).

  • The range excludes the value that the function cannot output, so the range is \( y eq -3 \).


For its inverse \( f^{-1}(x) = \frac{12 + 6x}{x + 3} \):

  • The domain excludes the value that makes its denominator zero, so the domain is \( x eq -3 \).

  • The range excludes the value the function cannot output, so the range is \( y eq 6 \).

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

Solve each rational inequality. Write each solution set in interval notation. $$\frac{-4}{1-x}<5$$

Simplify each power of i. $$\frac{1}{i^{-12}}$$

Solve each equation or inequality. $$|8 x+5|=0$$

Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\) (the opposition to current). These three quantities are related by the equation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation \(E=I Z\) for the remaining value. $$I=20+12 i, Z=10-5 i$$

Solve each equation or inequality. $$|4 x+3|>0$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

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.