Question

inverse of functions 6
this is the only question in this section.
question
find $f^{-1}(x)$ if $f(x) = \frac{x - 4}{7}$
answer attempt 1 out of 99
additional solution no solution
$f^{-1}(x) = \square$ submit answer

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = \frac{x - 4}{7} \). This is a standard first step in finding the inverse of a function, as it helps us work with \( y \) and \( x \) more easily.

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). So we get \( x = \frac{y - 4}{7} \). This is based on the definition of an inverse function, where if \( y = f(x) \), then \( x = f^{-1}(y) \), and we want to express \( y \) in terms of \( x \) for the inverse function.

Step3: Solve for \( y \)

First, multiply both sides of the equation \( x = \frac{y - 4}{7} \) by 7 to get rid of the denominator on the right side. This gives us \( 7x = y - 4 \). Then, add 4 to both sides of the equation to solve for \( y \). So \( y = 7x + 4 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

Now that we have solved for \( y \) in terms of \( x \), we replace \( y \) with \( f^{-1}(x) \) to get the inverse function. So \( f^{-1}(x) = 7x + 4 \).

Answer:

\( f^{-1}(x) = 7x + 4 \)