Problem 87 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I found the inverse of \(f(x)=5 x-4\) in my head: The reverse of multiplying by 5 and subtracting 4 is adding 4 and dividing by \(5,\) so \(f^{-1}(x)=\frac{x+4}{5}\).

Step-by-Step Solution

Answer

The statement made by the student does make sense, the inverse function is correctly calculated as \(f^{-1}(x) = \frac{x + 4}{5}\).

Step 1: Understand the concept of an inverse function

An inverse function is a function that undoes the work of the original function. In other words, it turns the output of the initial function back into the input. If the original function \(f(x)\) replaces an input \(x\) by an output \(5x - 4\), then the inverse function \(f^{-1}(x)\) should replace an input \(y = 5x - 4\) with its original related \(x\).

Step 2: Convert the original function to the inverse form

Let's say \(y = 5x - 4\). To find the inverse function, we need to solve for \(x\). That would include these steps: 1) Adding 4 to both sides gives \(y + 4 = 5x\). 2) Then we divide by 5, getting \(\frac{y + 4}{5} = x\). This equation can be simply restated as \(f^{-1}(y) = \frac{y + 4}{5}\) to emphasize that it is the inverse function.

Step 3: Validate the given statement

When we compare the function that we calculated, which is \(f^{-1}(y) = \frac{y + 4}{5}\), with the one stated by the student, which is \(f^{-1}(x) = \frac{x + 4}{5}\), we see that they match. Therefore, the statement of the student does make sense.