Question

if $g(x) = \frac{2}{3}x - 21$, find $g(g^{-1}(25))$.

Explanation:

Step1: Recall the property of inverse functions

For a function \( g \) and its inverse \( g^{-1} \), the composition \( g(g^{-1}(x)) = x \) for all \( x \) in the domain of \( g^{-1} \) (and range of \( g \)). This is because applying a function and then its inverse (or vice versa) undoes the operation, returning the original input.

Step2: Apply the property to the given expression

We need to find \( g(g^{-1}(25)) \). Using the property from Step 1, where \( x = 25 \), we have \( g(g^{-1}(25)) = 25 \). The intermediate step of finding \( g^{-1}(25) \) and then applying \( g \) to it is unnecessary because of the inverse function property. Even though we know the specific form of \( g(x)=\frac{2}{3}x - 21 \), we can use the general property of inverse functions to simplify the calculation.

Answer:

25