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Let's analyze and solve the given problem. The functions are:

\[
f(x) = 5e^x \quad \text{and} \quad g(x) = x^5
\]

We are tasked with finding the following values:

### (a) \( f(g(1)) \)
This means we first calculate \( g(1) \), then use that result as the input to \( f \).

\[
g(1) = 1^5 = 1
\]
Now, substitute into \( f \):
\[
f(1) = 5e^1 = 5e
\]
Thus, \( f(g(1)) = 5e \).

### (b) \( g(f(1)) \)
First, calculate \( f(1) \):
\[
f(1) = 5e^1 = 5e
\]
Now, use this in \( g \):
\[
g(5e) = (5e)^5 = 3125e^5
\]
Thus, \( g(f(1)) = 3125e^5 \).

### (c) \( f(g(x)) \)
We are asked to find the expression for \( f(g(x)) \), which is \( f(x^5) \):
\[
f(g(x)) = f(x^5) = 5e^{x^5}
\]
Thus, \( f(g(x)) = 5e^{x^5} \).

### (d) \( g(f(x)) \)
We are asked to find the expression for \( g(f(x)) \), which is \( g(5e^x) \):
\[
g(f(x)) = (5e^x)^5 = 3125e^{5x}
\]
Thus, \( g(f(x)) = 3125e^{5x} \).

### (e) \( f(t)g(t) \)
This is simply the product of \( f(t) \) and \( g(t) \):
\[
f(t) = 5e^t, \quad g(t) = t^5
\]
\[
f(t)g(t) = (5e^t)(t^5) = 5t^5e^t
\]
Thus, \( f(t)g(t) = 5t^5e^t \).

### Final answers:
- (a) \( f(g(1)) = 5e \)
- (b) \( g(f(1)) = 3125e^5 \)
- (c) \( f(g(x)) = 5e^{x^5} \)
- (d) \( g(f(x)) = 3125e^{5x} \)
- (e) \( f(t)g(t) = 5t^5e^t \)

Would you like more details or have any questions? 

Here are five related questions to expand on this:

1. What would be \( f(g(2)) \) if \( g(x) = x^4 \) instead?
2. How does the chain rule apply when differentiating \( f(g(x)) \)?
3. What is the inverse of \( f(x) = 5e^x \)?
4. How would the results change if \( g(x) \) were a different polynomial function, such as \( g(x) = x^2 \)?
5. How would you solve for \( x \) if \( f(x) = g(x) \)?

**Tip**: Understanding the composition of functions \( f(g(x)) \) or \( g(f(x)) \) helps in calculus, particularly when dealing with the chain rule for derivatives.

For the function f(x) = 5e^x and g(x) = x^5, find the following: (a) f(g(1)) (b) g(f(1)) (c) f(g(x)) (d) g(f(x)) (e) f(t)g(t).

Solve the function composition problem involving f(x) = 5e^x and g(x) = x^5. Learn how to find f(g(1)), g(f(1)), and other expressions through step-by-step solutions. Suitable for Grades 11-12 or early college-level students.

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