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Chapter 10: Problem 31

Determine whether each function is one-to-one. If it is, find the inverse. $$g(x)=(x+1)^{3}$$

Short Answer

Expert verified
The function \(g(x) = (x+1)^3\) is one-to-one, and its inverse is \(g^{-1}(x) = \sqrt[3]{x} - 1\).

Step by step solution

01

Determine if the function is one-to-one

A function is one-to-one if each output is produced by exactly one input. To check this for the function \(g(x) = (x+1)^3\), let's determine if it passes the Horizontal Line Test.
02

Horizontal Line Test

The Horizontal Line Test states that a function is one-to-one if every horizontal line intersects the graph of the function at most once. Given the nature of the cubic function \((x+1)^3\), each horizontal line will intersect its graph at most once. Thus, \(g(x) = (x+1)^3\) is one-to-one.
03

Express the function with y

To find the inverse of the function, start by expressing \(g(x)\) using \(y\). Set \(y = (x+1)^3\).
04

Solve for x

Next, solve the equation \(y = (x+1)^3\) for \(x\). Take the cube root of both sides to get \(\sqrt[3]{y} = x + 1\).
05

Isolate x

Subtract 1 from both sides to isolate \(x\): \(x = \sqrt[3]{y} - 1\).
06

Express the inverse function

Replace \(y\) with \(x\) to express the inverse function. Thus, the inverse function is \(g^{-1}(x) = \sqrt[3]{x} - 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-to-One Functions
To determine if a function is one-to-one, we need to check that each output is produced by just one input. This means no two different inputs should produce the same output.

Imagine you have a function like our example, \(g(x) = (x + 1)^{3}\). Each value of \(x\) maps to a unique \(g(x)\). This uniqueness is a key characteristic of one-to-one functions.

One helpful test to determine if a function is one-to-one is the Horizontal Line Test. This leads us to the next concept.
Horizontal Line Test
The Horizontal Line Test helps us determine if a function is one-to-one. Draw horizontal lines through the graph of the function. If each horizontal line intersects the graph at most one time, then the function is one-to-one.

For the function \(g(x) = (x + 1)^{3}\), each horizontal line will intersect the graph only once. This confirms that \(g(x)\) is indeed a one-to-one function. Once confirmed, we can move on to finding the inverse, which is detailed next.
Solving Equations to Find Inverse Functions
To find the inverse function, start by expressing the original function in terms of \(y\). For our example, set \(y = (x + 1)^{3}\).

Next, solve this equation for \(x\) by isolating it on one side of the equation.

1. Take the cube root of both sides: \[ \sqrt[3]{y} = x + 1 \]

2. Subtract 1 from both sides to isolate \(x\): \[ x = \sqrt[3]{y} - 1\]

Lastly, switch \(y\) back to \(x\) to express the inverse function. Thus, the inverse is \(g^{-1}(x) = \sqrt[3]{x} - 1\).

This process of finding an inverse is vital for understanding the relationship between functions and their inverses.

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Most popular questions from this chapter

Revenues of software publishers in the United States for the years \(2004-2016\) can be modeled by the function $$ S(x)=91.412 e^{0.05195 x} $$ where \(x=4\) represents \(2004, x=5\) represents \(2005,\) and so on, and \(S(x)\) is in billions of dollars. Approximate, to the nearest unit, revenue for \(2016 .\) (Data from U.S. Census Bureau.)

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} \frac{1}{9} $$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{-x}=\pi $$

Solve each equation. Give exact solutions. $$ \log _{4}(2 x+8)=2 $$

Solve each equation. Give exact solutions. $$ \log _{2} x+\log _{2}(x-7)=3 $$
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