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Chapter 8: Problem 70

Describe how to find the inverse of a one-to-one function.

Short Answer

Expert verified
To find the inverse of a one-to-one function, replace the function with \(y\), swap \(x\) and \(y\), and then solve for \(y\).

Step by step solution

01

Definition of One-to-One Function

A one-to-one function, also called an injective function, is a function where every element of the range corresponds with one and only one element of the domain.
02

Definition of Inverse Function

The inverse of a function flips the function around the line \(y = x\), so the output values of the function become the input values of the inverse and vice versa. The inverse function of \(f\) is often denoted as \(f^{-1}\).
03

Procedure to Find Inverse Function

To find the inverse of a one-to-one function, first replace the function name with \(y\). Then switch the roles of \(y\) and \(x\); that is, swap the positions of \(y\) and \(x\) in the equation. After, solve the equation for \(y\) to get the inverse function.
04

Example

For example, take the one-to-one function \(f(x) = 3x+2\). Replace \(f(x)\) with \(y\) to give the equation \(y = 3x + 2\). Swap \(x\) and \(y\) to get \(x = 3y + 2\). Finally, solve for \(y\) to get \(y = (x - 2) / 3\), which is the inverse function \(f^{-1}(x) = (x - 2) / 3\).

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The function \(f(x)=5\) is one-to-one.

Will help you prepare for the material covered in the first section of the next chapter. Solve: \(2-12 x=7(x-1)\).

Solve: \(12-2(3 x+1)=4 x-5\).

If \(f(x+y)=f(x)+f(y)\) and \(f(1)=3,\) find \(f(2), f(3)\) and \(f(4) .\) Is \(f(x+y)=f(x)+f(y)\) for all functions?

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=x+5$$
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