https://file.aiquickdraw.com/mathbot/file/1726074926951koqd0kum.jpg

Let's solve each part of the problem step by step:

### Part (a)
We are given the function:
\[ y = (7t^5 - 8)^6 \]

- **Inner function:** The expression inside the power is \( 7t^5 - 8 \).
  So, let \( u = 7t^5 - 8 \).

- **Outer function:** Now, \( y \) becomes a power function of \( u \), i.e.
  \[
  y = u^6
  \]

Thus, for part (a):
- \( u = 7t^5 - 8 \)
- \( y = u^6 \)

### Part (b)
We are given the function:
\[ P = 12e^{-0.3t} \]

- **Inner function:** The exponent part is \( -0.3t \).
  So, let \( u = -0.3t \).

- **Outer function:** \( P \) becomes an exponential function with base \( e \), i.e.
  \[
  P = 12e^u
  \]

Thus, for part (b):
- \( u = -0.3t \)
- \( P = 12e^u \)

### Part (c)
We are given the function:
\[ C = 11\ln(q^4 + 6) \]

- **Inner function:** The expression inside the logarithm is \( q^4 + 6 \).
  So, let \( u = q^4 + 6 \).

- **Outer function:** \( C \) becomes a logarithmic function, i.e.
  \[
  C = 11\ln(u)
  \]

Thus, for part (c):
- \( u = q^4 + 6 \)
- \( C = 11\ln(u) \)

---

Would you like further details or have any questions?

### Relative Questions:
1. What is the general process for expressing a function as a composition of two functions?
2. How can we differentiate a composite function using the chain rule?
3. How would the outer function change if \( y = (7t^5 - 8)^7 \) instead?
4. Can you express \( P = 12e^{0.3t} \) as a composition of functions, and what would change?
5. How does the choice of inner function affect the derivative of composite functions?

### Tip:
When identifying composite functions, always look for the "outer" function (e.g., a power, exponential, or logarithmic function) and the "inner" function (typically what's inside parentheses, exponents, etc.).

Express the given functions as compositions of two functions.

Learn how to express functions like y = (7t^5 - 8)^6, P = 12e^{-0.3t}, and C = 11ln(q^4 + 6) as compositions of two functions. This detailed guide covers power, exponential, and logarithmic functions, demonstrating how to identify inner and outer functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation