For each pair of functions, find ( f ◦ g) (x) and (g ◦ f) (x). 11. f(x) = 2x + 1, 12. f(x) = x2 + 3,
g(x) = x − 3 g(x) = 2x − 1
13. f(x) = −x2 − 8, g(x) = x2 − 1 14. f(x) = x + 2,
g(x) = x − 2 

To solve for \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we need to recall the definition of composition of functions. For two functions \( f(x) \) and \( g(x) \):

- \( (f \circ g)(x) = f(g(x)) \), meaning you substitute \( g(x) \) into \( f(x) \).
- \( (g \circ f)(x) = g(f(x)) \), meaning you substitute \( f(x) \) into \( g(x) \).

### Problem 11:
Given:
- \( f(x) = 2x + 1 \)
- \( g(x) = x - 3 \)

#### 1. Find \( (f \circ g)(x) = f(g(x)) \):
\[
f(g(x)) = f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5
\]

#### 2. Find \( (g \circ f)(x) = g(f(x)) \):
\[
g(f(x)) = g(2x + 1) = (2x + 1) - 3 = 2x - 2
\]

### Problem 12:
Given:
- \( f(x) = x^2 + 3 \)
- \( g(x) = 2x - 1 \)

#### 1. Find \( (f \circ g)(x) = f(g(x)) \):
\[
f(g(x)) = f(2x - 1) = (2x - 1)^2 + 3 = 4x^2 - 4x + 1 + 3 = 4x^2 - 4x + 4
\]

#### 2. Find \( (g \circ f)(x) = g(f(x)) \):
\[
g(f(x)) = g(x^2 + 3) = 2(x^2 + 3) - 1 = 2x^2 + 6 - 1 = 2x^2 + 5
\]

### Problem 13:
Given:
- \( f(x) = -x^2 - 8 \)
- \( g(x) = x^2 - 1 \)

#### 1. Find \( (f \circ g)(x) = f(g(x)) \):
\[
f(g(x)) = f(x^2 - 1) = -(x^2 - 1)^2 - 8 = -(x^4 - 2x^2 + 1) - 8 = -x^4 + 2x^2 - 1 - 8 = -x^4 + 2x^2 - 9
\]

#### 2. Find \( (g \circ f)(x) = g(f(x)) \):
\[
g(f(x)) = g(-x^2 - 8) = (-x^2 - 8)^2 - 1 = (x^4 + 16x^2 + 64) - 1 = x^4 + 16x^2 + 63
\]

### Problem 14:
Given:
- \( f(x) = x + 2 \)
- \( g(x) = x - 2 \)

#### 1. Find \( (f \circ g)(x) = f(g(x)) \):
\[
f(g(x)) = f(x - 2) = (x - 2) + 2 = x
\]

#### 2. Find \( (g \circ f)(x) = g(f(x)) \):
\[
g(f(x)) = g(x + 2) = (x + 2) - 2 = x
\]

### Summary:
- **Problem 11**: 
  - \( (f \circ g)(x) = 2x - 5 \)
  - \( (g \circ f)(x) = 2x - 2 \)
- **Problem 12**: 
  - \( (f \circ g)(x) = 4x^2 - 4x + 4 \)
  - \( (g \circ f)(x) = 2x^2 + 5 \)
- **Problem 13**: 
  - \( (f \circ g)(x) = -x^4 + 2x^2 - 9 \)
  - \( (g \circ f)(x) = x^4 + 16x^2 + 63 \)
- **Problem 14**: 
  - \( (f \circ g)(x) = x \)
  - \( (g \circ f)(x) = x \)

Would you like further explanation on any of these steps?

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Here are five related questions to test your understanding:
1. What is the composition \( (f \circ g)(x) \) if \( f(x) = 3x + 1 \) and \( g(x) = x^2 - 4 \)?
2. How would the result change for \( (g \circ f)(x) \) if \( f(x) = x^2 + 4 \) and \( g(x) = 2x + 1 \)?
3. Can the composition of two functions ever result in a constant value for all \( x \)?
4. What is the difference between \( f(g(x)) \) and \( g(f(x)) \) in terms of domain restrictions?
5. If \( f(x) = x^3 \) and \( g(x) = \frac{1}{x} \), what is \( (f \circ g)(x) \)?

**Tip**: Always ensure that the range of the inner function is within the domain of the outer function when composing functions.

For each pair of functions, find (f ◦ g)(x) and (g ◦ f)(x). 11. f(x) = 2x + 1, g(x) = x − 3 12. f(x) = x^2 + 3, g(x) = 2x − 1 13. f(x) = −x^2 − 8, g(x) = x^2 − 1 14. f(x) = x + 2, g(x) = x − 2

Learn how to solve for function compositions (f ◦ g)(x) and (g ◦ f)(x) with polynomial functions. This step-by-step guide explores algebraic concepts and key formulas. Suitable for students in Grades 10-12.

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