For 𝑓(𝑥) = 𝑥2 −1 and 𝑔(𝑥) = 3𝑥 +1, find  
a) 𝑓∘𝑔 
b) 𝑔∘𝑓 

Given two functions \( f(x) = x^2 - 1 \) and \( g(x) = 3x + 1 \), we are tasked with finding:

### a) \( f \circ g \) (also written as \( f(g(x)) \)):
To find \( f \circ g \), we substitute \( g(x) \) into \( f(x) \).

\[
f(g(x)) = f(3x + 1) 
\]

Now, substitute \( 3x + 1 \) for \( x \) in \( f(x) \):

\[
f(3x + 1) = (3x + 1)^2 - 1
\]

First, expand \( (3x + 1)^2 \):

\[
(3x + 1)^2 = 9x^2 + 6x + 1
\]

Now subtract 1:

\[
f(g(x)) = 9x^2 + 6x + 1 - 1 = 9x^2 + 6x
\]

Thus,

\[
f \circ g = 9x^2 + 6x
\]

### b) \( g \circ f \) (also written as \( g(f(x)) \)):
To find \( g \circ f \), we substitute \( f(x) \) into \( g(x) \).

\[
g(f(x)) = g(x^2 - 1)
\]

Now, substitute \( x^2 - 1 \) for \( x \) in \( g(x) \):

\[
g(x^2 - 1) = 3(x^2 - 1) + 1
\]

Expand:

\[
g(x^2 - 1) = 3x^2 - 3 + 1 = 3x^2 - 2
\]

Thus,

\[
g \circ f = 3x^2 - 2
\]

### Final Results:
a) \( f \circ g = 9x^2 + 6x \)

b) \( g \circ f = 3x^2 - 2 \)

Would you like further details or explanations on any of the steps?

Here are 5 related questions:
1. What is the domain of \( f \circ g \)?
2. Can you reverse the order of function composition, i.e., does \( f \circ g = g \circ f \)?
3. How would you compute \( f \circ f(x) \) and \( g \circ g(x) \)?
4. How can you graph \( f \circ g \) and \( g \circ f \) on the same coordinate plane?
5. What happens if \( g(x) \) is not linear—how does that change \( f \circ g \)?

**Tip**: When composing functions, always substitute one function into the other carefully, ensuring all operations are applied correctly.

For f(x) = x^2 − 1 and g(x) = 3x + 1, find a) f∘g b) g∘f

Learn how to solve the function compositions f∘g and g∘f for the functions f(x) = x^2 − 1 and g(x) = 3x + 1. This detailed step-by-step solution explores function composition and algebraic concepts, ideal for students in Grades 9-11.

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