For the functions f(x) = e ^ (8x) and g(x) = x ^ 7 find the following. NOTE : Enter the exact answers (a) (b) ( ) f(g(1)) =; g(f(1)) =; f(g(x)) =; g(f(x)) =; f(t) * g(t) = (d) ( e)

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.

For the functions f(x) = e ^ (8x) and g(x) = x ^ 7 find the following. NOTE: Enter the exact answers (a) f(g(1)) =; g(f(1)) =; f(g(x)) =; g(f(x)) =; f(t) * g(t) =

Explore how to solve function compositions and products for f(x) = e^(8x) and g(x) = x^7. Learn to find exact expressions for f(g(1)), g(f(1)), f(g(x)), g(f(x)), and the product f(t) * g(t). Suitable for advanced high school or college-level students.

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