Let's work through each part step by step:

Given functions:

• $f(x) = e^{8x}$
• $g(x) = x^7$

(a) Finding $f(g(1))$ and $g(f(1))$:

1. $f(g(1))$:

   • First, calculate $g(1)$: $g(1) = 1^7 = 1$
   • Now substitute $g(1) = 1$ into $f(x)$: $f(g(1)) = f(1) = e^{8(1)} = e^8$ So, $f(g(1)) = e^8$.

2. $g(f(1))$:

   • First, calculate $f(1)$: $f(1) = e^{8(1)} = e^8$
   • Now substitute $f(1) = e^8$ into $g(x)$: $g(f(1)) = (e^8)^7 = e^{56}$ So, $g(f(1)) = e^{56}$.

(b) Finding $f(g(x))$ and $g(f(x))$:

1. $f(g(x))$:

   • Substitute $g(x) = x^7$ into $f(x)$: $f(g(x)) = f(x^7) = e^{8(x^7)} = e^{8x^7}$ So, $f(g(x)) = e^{8x^7}$.

2. $g(f(x))$:

   • Substitute $f(x) = e^{8x}$ into $g(x)$: $g(f(x)) = (e^{8x})^7 = e^{56x}$ So, $g(f(x)) = e^{56x}$.

(c) Finding $f(t) \cdot g(t)$:

$f(t) \cdot g(t) = e^{8t} \cdot t^7$

This is the exact expression for the product of $f(t)$ and $g(t)$.

Summary:

• $f(g(1)) = e^8$
• $g(f(1)) = e^{56}$
• $f(g(x)) = e^{8x^7}$
• $g(f(x)) = e^{56x}$
• $f(t) \cdot g(t) = e^{8t} \cdot t^7$

Let me know if you want more details or have any questions.

Here are 5 related questions to deepen understanding:

1. What is the derivative of $f(x) = e^{8x}$?
2. How does the composition of functions affect their domain and range?
3. If $f(x)$ is an exponential function and $g(x)$ is a polynomial, what can be said about their growth rates?
4. What would happen if $g(x) = x^n$ where $n$ is an even number instead of 7?
5. How would you integrate the product $f(t) \cdot g(t)$?

Tip: When working with compositions of functions, it’s essential to keep track of which function is applied first.