https://file.aiquickdraw.com/mathbot/file/1726524578748culmflxv.jpg

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 \( f(g(x)) \).

For the function f(x) = 4e^x and g(x) = x^7, find the following: (a) f(g(1)), (b) g(f(1)), (c) f(g(x)), (d) g(f(x)), (e) f(t)g(t).

Explore the composition of functions involving exponential and polynomial functions. Solve problems such as f(g(1)), g(f(1)), and more, using the given functions f(x) = 4e^x and g(x) = x^7. This topic involves function composition and properties of exponentials and polynomials.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation