What is the inverse of f(x) = 4x + 5?

The inverse of \( f(x) = 4x + 5 \) is \( f^{-1}(x) = \frac{x - 5}{4} \).

To find the inverse of a function, we need to swap the roles of \( x \) and \( y \) and then solve for \( y \). Let's start with the given function \( f(x) = 4x + 5 \). First, we replace \( f(x) \) with \( y \) to make it easier to work with:

\[ y = 4x + 5 \]

Next, we swap \( x \) and \( y \):

\[ x = 4y + 5 \]

Now, we need to solve this equation for \( y \). Start by isolating \( y \) on one side of the equation. Subtract 5 from both sides:

\[ x - 5 = 4y \]

Then, divide both sides by 4:

\[ y = \frac{x - 5}{4} \]

So, the inverse function \( f^{-1}(x) \) is:

\[ f^{-1}(x) = \frac{x - 5}{4} \]

This means that if you have a value for \( x \) in the original function \( f(x) \), you can use the inverse function \( f^{-1}(x) \) to find the original input value. For example, if \( f(x) = 13 \), you can find \( x \) by using the inverse function:

\[ x = f^{-1}(13) = \frac{13 - 5}{4} = 2 \]

This process of finding the inverse is useful in many areas of mathematics and helps to understand how functions and their inverses relate to each other.