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Chapter 3: Problem 37

Find the inverse function of \(f\) $$ f(x)=2 x+1 $$

Short Answer

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

Step by step solution

01

Set Up Function Equation

Start by writing down the function equation you're given: \(y = 2x + 1\). The goal is to find the inverse, which means solving this equation for \(x\) in terms of \(y\).
02

Swap Variables

To find the inverse function, swap the roles of \(x\) and \(y\). This translates to \(x = 2y + 1\). Now the solution will isolate \(y\) in terms of \(x\).
03

Isolate the Variable

Subtract 1 from both sides of the equation: \(x - 1 = 2y\). This step begins isolating the \(y\) variable on one side of the equation.
04

Solve for the Inverse

Divide both sides by 2 to solve for \(y\): \(y = \frac{x - 1}{2}\).
05

Identify the Inverse Function

Recognize that \(y = \frac{x - 1}{2}\) is the inverse function \(f^{-1}(x)\). Therefore, the inverse of the original function is \(f^{-1}(x) = \frac{x - 1}{2}\).

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Key Concepts

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

Function Equation
When dealing with functions, the function equation is the starting point. In our given problem, the function equation is expressed as \( f(x) = 2x + 1 \). This notation shows us how the function manipulates the input \(x\) to produce the output. Understanding the function equation is crucial because it provides a mathematical model of how inputs and outputs are related. Every function equation follows a specific pattern or rule which tells you exactly how to transform an input to get an output.
Swap Variables
Once you have the function equation, the next step to find the inverse is to swap the variables. In this context, it means changing the roles of \(x\) and \(y\) in the equation. Initially, we have \( y = 2x + 1 \). For inversion, we rewrite the equation as \( x = 2y + 1 \).Swapping variables is essential in finding the inverse because the inverse function essentially reverses the original function's operation. It helps you see how to go back from the output \(y\) to find the original input \(x\), essentially tracing the steps of the function in reverse.
Isolate Variable
After swapping variables, the next step is to isolate the variable we are solving for, which in this case is \(y\). It involves rearranging the equation, \( x = 2y + 1 \), such that \(y\) is by itself on one side. Begin by subtracting 1 from both sides: \( x - 1 = 2y \).This step is critical as it simplifies the equation down to a form where the desired variable is isolated. It prepares the equation for the final step of explicitly solving for \(y\). The process of isolation makes it easier to identify what arithmetic manipulations are needed to get to the end result.
Solve for Inverse
Finally, to solve for the inverse, divide both sides of the equation \( x - 1 = 2y \) by 2. This effectively isolates \(y\), resulting in the equation \( y = \frac{x - 1}{2} \). Consequently, the inverse function is determined as \( f^{-1}(x) = \frac{x - 1}{2} \).Finding the inverse function demonstrates how you can transform an output back into an input, essentially running the function in reverse. The notation \( f^{-1}(x) \) expresses that we now have a function such that when you plug in the output from the original function, you retrieve the original input. It is a powerful tool, widely applicable in solving mathematical problems dealing with reversible processes.

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Most popular questions from this chapter

Pizza Cost Marcello's Pizza charges a base price of \(\$ 7\) for a large pizza plus \(\$ 2\) for each topping. Thus, if you order a large pizza with \(x\) toppings, the price of your pizza is given by the function \(f(x)=7+2 x .\) Find \(f^{-1} .\) What does the function \(f^{-1}\) represent?

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. $$ f(x)=|x|-|x-6| $$

Finding an Inverse "in Your Head" In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 6 we saw that the inverse of $$ f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3} $$ because the "reverse" of "Multiply by 3 and subtract \(2^{\prime \prime}\) is "Add 2 and divide by 3 ." Use the same procedure to find the inverse of the following functions. $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{2 x+1}{5}} & {\text { (b) } f(x)=3-\frac{1}{x}} \\ {\text { (c) } f(x)=\sqrt{x^{3}+2}} & {\text { (d) } f(x)=(2 x-5)^{3}}\end{array} $$ Now consider another function: $$ f(x)=x^{3}+2 x+6 $$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

Exchange Rates The relative value of currencies fluctuates every day. When this problem was written, one Canadian dollar was worth 1.0573 U.S. dollar. (a) Find a function \(f\) that gives the U.S. dollar value \(f(x)\) of \(X\) Canadian dollars. (b) Find \(f^{-1} .\) What does \(f^{-1}\) represent? (c) How much Canadian money would \(\$ 12,250\) in U.S. currency be worth?

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{x}{x+1}, \quad g(x)=2 x-1 $$
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