Find 
f ∘ g
 and 
g ∘ f.
f(x) = 
3

x − 9
,    g(x) = x3 + 1
(a)    
f ∘ g


To find \( f \circ g \), we need to evaluate \( f(g(x)) \), which means we substitute \( g(x) \) into \( f(x) \).

Given the functions:
- \( f(x) = 3x - 9 \)
- \( g(x) = x^3 + 1 \)

### Step 1: Substitute \( g(x) \) into \( f(x) \)
\[
f(g(x)) = 3(g(x)) - 9
\]
Since \( g(x) = x^3 + 1 \), substitute this into \( f(g(x)) \):
\[
f(g(x)) = 3(x^3 + 1) - 9
\]

### Step 2: Simplify the expression
\[
f(g(x)) = 3x^3 + 3 - 9
\]
\[
f(g(x)) = 3x^3 - 6
\]

Thus, \( f \circ g = 3x^3 - 6 \).

Would you like to proceed with \( g \circ f \)? Let me know if you want more details or have any questions.

Here are 5 questions related to the problem:
1. What is the domain of the function \( f \circ g \)?
2. How does the composition of functions affect their range?
3. Can the order of composition \( f \circ g \) change the result compared to \( g \circ f \)?
4. What happens if \( g(x) \) is a non-polynomial function?
5. How do we interpret function composition graphically?

**Tip**: Always pay attention to the domain of both functions when performing compositions to avoid undefined values!

Find f ∘ g and g ∘ f. f(x) = 3x − 9, g(x) = x^3 + 1 (a) f ∘ g

Learn how to find the composition of functions f ∘ g and g ∘ f where f(x) = 3x - 9 and g(x) = x^3 + 1. This solution includes step-by-step explanations for substituting and simplifying expressions using algebraic methods. Suitable for students in Grades 10-12.

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