Identify the inside function,
u = g(x),
and the outside function,
y = f(u).
y = (x2 − 7x + 8)^3

To identify the inside function \( u = g(x) \) and the outside function \( y = f(u) \) for the equation \( y = (x^2 - 7x + 8)^3 \), follow these steps:

### 1. **Inside Function (\( u = g(x) \)):**
The **inside function** is the expression inside the parentheses. Here, it is \( x^2 - 7x + 8 \). So, we have:
\[
u = g(x) = x^2 - 7x + 8
\]

### 2. **Outside Function (\( y = f(u) \)):**
The **outside function** is the function that is applied to the result of the inside function. In this case, the outer function involves cubing the expression inside the parentheses. So, we have:
\[
y = f(u) = u^3
\]

### Final Answer:
- Inside function: \( u = g(x) = x^2 - 7x + 8 \)
- Outside function: \( y = f(u) = u^3 \)

Let me know if you'd like further clarification or more details! Here are a few related questions to explore:

1. How do you apply the chain rule to differentiate a composition of functions like this?
2. What is the derivative of the function \( y = (x^2 - 7x + 8)^3 \)?
3. How does the chain rule work when dealing with higher powers or nested functions?
4. What would happen if the exponent in the outside function was a fraction, like \( (x^2 - 7x + 8)^{1/2} \)?
5. Can you identify the inside and outside functions in trigonometric compositions, such as \( y = \sin(x^2 - 7x + 8) \)?

**Tip**: Always look for the expression that is being operated on first (inside), and then identify the operation performed on that expression (outside).

Identify the inside function, u = g(x), and the outside function, y = f(u). y = (x^2 − 7x + 8)^3

Learn how to identify the inside and outside functions in compositions such as y = (x^2 − 7x + 8)^3. This problem covers function composition, helping students recognize inner and outer functions for differentiation using the chain rule. Suitable for students in Grades 9-11.

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