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Chapter 5: Problem 83

Describe how to find the inverse function of a one-to-one function given by an equation in \(x\) and \(y .\) Give an example.

Short Answer

Expert verified
The steps to find the inverse function of a one-to-one function include defining what a one-to-one function is, swapping the variables of the function, solving for the other variable, verifying the inverse function through the composite function test, and demonstrating the process with an example. An example inverse function for \(y = 2x + 3\) is \(f^{-1}(x) = (x - 3) / 2\).

Step by step solution

01

Define One-to-One function

A one-to-one function, also known as an injective function, is a function where each element of the range corresponds exclusively to one element of the domain. For example, for function \(f(x)\), if \(f(a) = f(b)\), then \(a = b\). This means that if two different inputs give the same output, those inputs must be the same.
02

Swap the variables

To find the inverse function, the first step involves swapping the places of \(x\) and \(y\). Let's take \(y = f(x)\) as an example of a one-to-one function. By swapping, we get \(x = f(y)\). This step is done because in an inverse function, inputs become outputs and outputs become inputs.
03

Solve for the other variable

The next step is to solve for the other variable. This typically involves rearranging the equation to isolate \(y\). The process to do this will depend on the original equation. Assume the example equation \(x = 2y + 3\). We will subtract 3 from both sides to get \(x - 3 = 2y\) then divide by 2 to get \(y = (x - 3) / 2\). This is the inverse of the original function.
04

Verify the inverse function

It's always a good idea to verify if you have correctly obtained the inverse function. For this, you can use the composite function test. If you evaluate \(f(|f^{-1}(x)|)\) or \(f^{-1}(f(x))\), in both cases you should get \(x\), since applying a function to its inverse (and vice versa) should give you the original value. You can use a specific value for \(x\) to check.
05

Demonstrate with example

Let's consider a one-to-one function \(y = 2x + 3\). Swap the variables, we get \(x = 2y + 3\). Solve for \(y\), we get \(y = (x - 3) / 2\). This is the inverse function, which is denoted by \(f^{-1}(x) = (x - 3) / 2\). Verify by substituting \(x\) in both functions, we should get the same \(x\) value back.

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