Let's determine the inverse functions for $f(x) = e^x$ and $g(x) = 5x$, and then find the expression for $g(f^{-1}(x))$.

1. Finding $f^{-1}(x)$ (the inverse of $f(x) = e^x$):

To find the inverse function $f^{-1}(x)$, we need to solve the equation $y = e^x$ for $x$.

$y = e^x$

Taking the natural logarithm (ln) on both sides:

$\ln(y) = x$

So, the inverse function is:

$f^{-1}(x) = \ln(x)$

2. Finding $g^{-1}(x)$ (the inverse of $g(x) = 5x$):

To find the inverse function $g^{-1}(x)$, we solve the equation $y = 5x$ for $x$.

$y = 5x$

Dividing both sides by 5:

$x = \frac{y}{5}$

So, the inverse function is:

$g^{-1}(x) = \frac{x}{5}$

3. Finding $g(f^{-1}(x))$:

We need to compute $g(f^{-1}(x))$.

First, substitute $f^{-1}(x) = \ln(x)$ into $g(x) = 5x$:

$g(f^{-1}(x)) = g(\ln(x)) = 5 \cdot \ln(x)$

So, the expression is:

$g(f^{-1}(x)) = 5\ln(x)$

Summary of the results:

• $f^{-1}(x) = \ln(x)$
• $g^{-1}(x) = \frac{x}{5}$
• $g(f^{-1}(x)) = 5\ln(x)$

Would you like more details or have any questions about these calculations?

Here are some related questions to consider:

1. What is the domain of the inverse function $f^{-1}(x) = \ln(x)$?
2. How would the expressions change if $f(x) = e^{2x}$ instead?
3. Can you determine $f(g^{-1}(x))$ and how it relates to $x$?
4. What are the properties of the compositions $f(f^{-1}(x))$ and $g(g^{-1}(x))$?
5. How do these inverses and their compositions affect the graphical interpretation of the functions?

Tip: When finding inverse functions, remember that applying the original function and its inverse in succession will return you to your starting value, i.e., $f(f^{-1}(x)) = x$ and $g(g^{-1}(x)) = x$.