Question

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

The inverse function of a function \(y = f(x)\) can be found by switching \(x\) and \(y\) in the equation and solving for \(y\). For example, the inverse function of \(y = 2x + 3\) is \(y = (x - 3)/2\).

Step-by-step solution

Verify the function is one-to-one

A function is one-to-one if every element in the range corresponds to exactly one element in the domain. This can be verified by the horizontal line test on its graph. If every horizontal line intersects the graph at most once, then the function is one-to-one and its inverse exists. For the purpose of this problem, let's proceed under the assumption that the given function is one-to-one.

Swap \(x\) and \(y\)

The next step is to replace each \(x\) with \(y\) and each \(y\) with \(x\) in the equation. The purpose of this step is because the inverse function undoes the work of the original function. If the original function converts \(x\) to \(y\), then the inverse function should convert \(y\) back into \(x\).

Solve for \(y\)

Now, solve the equation from Step 2 for \(y\). The resulting equation will be the inverse function.

Example

Consider the function \(y = 2x + 3\). Completing Step 2 gives: \(x = 2y + 3\). Now, solve this for \(y\): Subtract 3 from both sides to get \(x - 3 = 2y\), then divide both sides by 2 to find \(y = (x - 3)/2\). Thus, the inverse function is \(y = (x - 3)/2\).