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Chapter 12: Problem 13

If the function is one-to-one, find its inverse. \(f(x)=2 x+4\)

Short Answer

Expert verified
\(f^{-1}(x) = \frac{x - 4}{2}\)

Step by step solution

01

- Understand the given function

The given function is a linear function: \(f(x) = 2x + 4\). A linear function of this form is always one-to-one, which means it has a unique inverse.
02

- Replace f(x) with y

Write the function in terms of \(y\) instead of \(f(x)\): \(y = 2x + 4\).
03

- Swap x and y

To find the inverse, interchange \(x\) and \(y\): \(x = 2y + 4\).
04

- Solve for y

Isolate \(y\) on one side to find the inverse function: \(x = 2y + 4\). Subtract 4 from both sides: \(x - 4 = 2y\). Divide both sides by 2: \(y = \frac{x - 4}{2}\).
05

- Replace y with f^{-1}(x)

Replace \(y\) with \(f^{-1}(x)\): \(f^{-1}(x) = \frac{x - 4}{2}\). The inverse function is: \(f^{-1}(x) = \frac{x - 4}{2}\).

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Key Concepts

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

One-to-one Function
A one-to-one function, also known as an injective function, is a function where each input has a unique output. This means that if you have two different inputs, they should always map to two different outputs.
If a function is one-to-one, it is possible to find its inverse.
In the given exercise, the function \(f(x) = 2x + 4\) is one-to-one because it is a linear function with a non-zero slope.
This ensures that every input \(x\) maps to a unique output \(f(x)\).
This property is crucial for finding an inverse function because a non-unique mapping would complicate, or even make impossible, the determination of an inverse.
Linear Function
A linear function is a function that creates a straight line when graphed.
These functions take the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In the exercise, the function is given as \(f(x) = 2x + 4\).
Here, \(m = 2\) and \(b = 4\).
Linear functions are simple to work with due to their predictable and straightforward nature.
This makes them excellent candidates for exploring fundamental concepts like inverse functions.
Because linear functions with non-zero slopes are always one-to-one, they always have an inverse that also takes a linear form.
Solving Equations
Solving equations is a process of finding the value of a variable that makes an equation true.
In the steps given in the exercise, solving for the inverse function involves manipulating the equation to isolate the variable \(y\).
Here鈥檚 how it鈥檚 done:
  • Original function: \(f(x) = 2x + 4\)
  • Replace \(f(x)\) with \(y\): \(y = 2x + 4\)
  • Swap \(x\) and \(y\): \(x = 2y + 4\)
  • Solve for \(y\):
    - First, subtract 4: \(x - 4 = 2y\)
    - Then, divide by 2: \(y = \frac{x - 4}{2}\)
By isolating \(y\), we鈥檝e determined that \(f^{-1}(x) = \frac{x - 4}{2}\).
Understanding how to manipulate equations to isolate variables is essential for solving real-life problems and finding inverses of functions.

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Most popular questions from this chapter

Find each logarithm. Give approximations to four decimal places. \(\ln 8.32\)

To solve the equation \(5^{x}=7,\) we must find the exponent to which 5 must be raised in order to obtain \(7 .\) This is \(\log _{5} 7\) (a) Use the change-of-base rule and your calculator to find \(\log _{5} 7\). (b) Raise 5 to the number you found in part (a). What is your result? (c) Using as many decimal places as your calculator gives, write the solution set of \(5^{x}=7\) (Equations of this type will be studied in more detail in Section 12.6.)

Find each logarithm. Give approximations to four decimal places. \(\log 457.2\)

Find each logarithm. Give approximations to four decimal places. \(\ln 7.84\)

Let \(f(x)=2^{x} .\) We will see in the next section that this function is one- toone. Find each value, always working part (a) before part \((b)\). (a) \(f(3)\) (b) \(f^{-1}(8)\)
See all solutions

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