From the image, we need to solve the given function problems for $f(x) = 4e^x$ and $g(x) = x^7$.

Let's work through each part:

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

• $g(1) = 1^7 = 1$
• Now, $f(g(1)) = f(1) = 4e^1 = 4e$

(b) $g(f(1))$:

• $f(1) = 4e^1 = 4e$
• Now, $g(f(1)) = g(4e) = (4e)^7 = 4^7 e^7 = 4^7 e^7$

(c) $f(g(x))$:

• $g(x) = x^7$
• Now, $f(g(x)) = f(x^7) = 4e^{x^7}$

(d) $g(f(x))$:

• $f(x) = 4e^x$
• Now, $g(f(x)) = g(4e^x) = (4e^x)^7 = 4^7 (e^x)^7 = 4^7 e^{7x}$

(e) $f(t)g(t)$:

• $f(t) = 4e^t$
• $g(t) = t^7$
• Now, $f(t)g(t) = 4e^t \cdot t^7 = 4t^7 e^t$

Let me know if you need further clarifications or more details on these computations!

Here are some related questions:

1. What happens if we use a different base for $g(x)$ instead of powers of 7?
2. How would the answer change if $f(x)$ was a polynomial function instead of exponential?
3. How does the composition of two functions affect the overall growth rate?
4. What are the derivatives of $f(g(x))$ and $g(f(x))$?
5. Can these functions be inverted easily? What would $f^{-1}(x)$ and $g^{-1}(x)$ look like?

Tip: Always apply the chain rule when differentiating composite functions like $f(g(x))$.