Problem 11 - College Algebra

Question

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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$ f(x)=x+3 $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = x + 3\) is \(f^{-1}(x) = x - 3\). This has been verified using the definition of inverse functions.

Step 1: Find the Equation for the Inverse Function

The given function is \(f(x) = x + 3\). To find the inverse function, \(f^{-1}(x)\), replace \(f(x)\) with \(y\). So, initially we have \(y = x + 3\). Now, we can swap \(x\) and \(y\) to find the inverse function. Therefore, \(x = y + 3\). Solving for \(y\) (which is essentially \(f^{-1}(x)\)) gives \(f^{-1}(x) = x - 3\).

Step 2: Verifying the Equation for the Inverse Function

Now, we can verify the correctness of the inverse by checking if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). Substituting \(x-3\) for \(f^{-1}(x)\) in \(f(f^{-1}(x))\) gives \(f(x-3) = (x - 3) + 3 = x\). Substituting \(x+3\) for \(f(x)\) in \(f^{-1}(f(x))\) gives \(f^{-1}(x + 3) = (x + 3) - 3 = x\). Therefore, both equations hold true verifying that \(f^{-1}(x) = x - 3\) is indeed the correct inverse function for \(f(x) = x + 3\).

Key Concepts

One-to-One Functions

Understanding the concept of one-to-one functions—or bijections—is crucial when working with inverse functions. A one-to-one function is a type of mathematical function that has a unique output for each unique input. In simpler terms, every element in the domain of the function is paired with a distinct element in its range.

This unique pairing means that no two different domain values will produce the same range value—a critical requirement for a function to have an inverse. We can visualize this with a 'horizontal line test': draw a horizontal line across the graph of the function. If the line intersects the graph at exactly one point for every possible position of the line, the function is one-to-one.

To illustrate with an exercise, if we have a function like \(f(x) = x + 3\), we can clearly see that for every value of \(x\), the output will be unique. Thus, the function passes the horizontal line test and is confirmed to be one-to-one, making it valid to proceed with finding its inverse.

Equation Solving

Equation solving is a foundational skill in mathematics and is especially important when finding the inverse of a function. The basic principle involves manipulating an equation to isolate the variable of interest, typically on one side of the equation, with the goal of finding its value.

In the context of inverse functions, solving for the equation of the inverse involves a few steps. Starting by expressing the function as \(y = f(x)\), you then swap the roles of \(x\) and \(y\) to express the inverse relationship. The next step is to solve for \(y\), which will now represent the inverse function \(f^{-1}(x)\).

For the function \(f(x) = x + 3\), the first step is to let \(y = x + 3\). Upon swapping \(x\) and \(y\), we obtain \(x = y + 3\), which can be solved easily by subtracting 3 from both sides, yielding \(y = x - 3\). This new equation represents \(f^{-1}(x)\), our inverse function.

Function Composition

Function composition is the process of applying one function to the results of another function. When the functions involved are inverses of each other, composing them should return the original input. In mathematical terms, if you have a function \(f\) and its inverse \(f^{-1}\), then for any \(x\) in the domain of \(f\), the compositions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) should both equal \(x\).

Verifying an inverse function's correctness involves composing the original function with its proposed inverse and checking whether or not the identity \(x\) is the result. For the function \(f(x) = x + 3\) with proposed inverse \(f^{-1}(x) = x - 3\), we should check both \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\).

In our example, substituting \(f^{-1}(x)\) into \(f\) gives us \(f(x - 3) = (x - 3) + 3 = x\), which shows the composition returns the original input. Similarly, substituting \(f(x)\) into \(f^{-1}\) gives us \(f^{-1}(x + 3) = (x + 3) - 3 = x\), confirming the inverse relationship.