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Chapter 12: Problem 37

Each function is one-to-one. Find its inverse. $$ f(x)=\frac{4 x-2}{x+5}, \quad x \neq-5 $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{-2 - 5x}{x - 4} \).

Step by step solution

01

Replace f(x) with y

To find the inverse, start by replacing the function notation. Write the given function as: \[ y = \frac{4x - 2}{x + 5} \]
02

Swap x and y

To find the inverse of the function, swap the variables x and y. This gives the equation: \[ x = \frac{4y - 2}{y + 5} \]
03

Solve for y

Next, solve this equation for y. Start by multiplying both sides by \( y + 5 \) to eliminate the fraction: \[ x(y + 5) = 4y - 2 \] Expand and rearrange the terms to isolate y: \[ xy + 5x = 4y - 2 \] \[ xy - 4y = -2 - 5x \] Factor out y from the left side: \[ y(x - 4) = -2 - 5x \] Finally, solve for y: \[ y = \frac{-2 - 5x}{x - 4} \]
04

Write the inverse function

After solving for y, replace y with \( f^{-1}(x) \): \[ f^{-1}(x) = \frac{-2 - 5x}{x - 4} \]

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Key Concepts

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

one-to-one functions
A one-to-one function is a type of function where each output is linked to a unique input. This means no two different inputs produce the same output. Identifying these functions is critical when finding inverses because only one-to-one functions have inverses that are also functions.
To determine if a function is one-to-one, you can use the horizontal line test. If any horizontal line crosses the graph of the function at most once, then the function is one-to-one.
  • Example: The function \( f(x) = 2x + 3 \) is one-to-one because every value of \( x \) maps to a unique \( y \), and no \( y \) repeats for different \( x \).
The function given in our problem, \( f(x) = \frac{4x - 2}{x + 5} \), is also one-to-one, making it possible to find its inverse.
function notation
Function notation is a way to denote functions in a simple, consistent manner. It usually involves a function name like \( f \) and an input like \( x \). For instance, \( f(x) = \frac{4x - 2}{x + 5} \) indicates that \( x \) is the variable, and applying function \( f \) to \( x \) produces the output.
  • When finding an inverse, we often start by rewriting \( f(x) \) as \( y \) for simplicity. This turns \( f(x) = \frac{4x - 2}{x + 5} \) into \( y = \frac{4x - 2}{x + 5} \).
In this notation, the inverse of the function \( f \) is symbolized as \( f^{-1} \). This notation is useful because it clearly indicates that we are working with an inverse, helping to avoid confusion as we solve the equation.
solving equations
Solving equations is the process of finding the values of variables that make the equations true. For inverse functions, this often involves multiple steps, including algebraic manipulation.
Let's go through solving the given equation step by step:
  • First, swap \( x \) and \( y \) in the equation: \( x = \frac{4y - 2}{y + 5} \).
  • Next, eliminate the fraction by multiplying both sides by \( y + 5 \): \( x(y + 5) = 4y - 2 \).
  • Then, expand and rearrange to isolate terms involving \( y \): \( xy + 5x = 4y - 2 \) becomes \( xy - 4y = -2 - 5x \).
  • Factor \( y \) out of the left side: \( y(x - 4) = -2 - 5x \).
  • Finally, solve for \( y \): \( y = \frac{-2 - 5x}{x - 4} \).
Now that we have isolated \( y \), we replace \( y \) with \( f^{-1}(x) \) to write the inverse function: \( f^{-1}(x) = \frac{-2 - 5x}{x - 4} \). Solving these types of equations requires patience and practice, but mastering these skills will make finding inverses much easier.

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Most popular questions from this chapter

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\left(\log _{a} r-\log _{a} s\right)+3 \log _{a} t$$

Use the special properties of logarithms to evaluate each expression. \(\log _{3} 3\)

Solve each equation. \(x=\log _{125} 5\)

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{6} \sqrt[3]{5}\)

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=-4 x $$
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