Consider the functions *f* and *g* that are defined for *xx*∈*R* by *xf*(*x*)=*ex* and *xg*(*x*)=5*x*.

Determine the expression of x*f*−1(*x*), x*g*−1(*x*) and x()*gf*−1(*x*)

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 \( g(g^{-1}(x)) = x \).

Explore how to find inverse functions for exponential and linear functions, including step-by-step examples. Learn how to calculate expressions like g(f^{-1}(x)) and more. Suitable for advanced high school students and college-level mathematics.

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