14. f(g(2))

15. f(f(1))

16. (f ∘ g)(6)

17. (f ∘ f)(5)

Find the functions f ∘ g, g ∘ f, f ∘ f, and g ∘ g and their domains.

18. f(x) = 2x + 3, g(x) = 4x - 1
the given is f(x) = 2x -3 and g(×) =4 - x²

Let's break this down step by step, focusing on each part of the question.

### 14. \( f(g(2)) \)

Given:
- \( f(x) = 2x - 3 \)
- \( g(x) = 4 - x^2 \)

We first need to evaluate \( g(2) \) and then substitute the result into \( f(x) \):

1. \( g(2) = 4 - (2)^2 = 4 - 4 = 0 \)
2. \( f(g(2)) = f(0) = 2(0) - 3 = -3 \)

Thus, \( f(g(2)) = -3 \).

---

### 15. \( f(f(1)) \)

We need to evaluate \( f(1) \) first and then substitute that value back into \( f(x) \).

1. \( f(1) = 2(1) - 3 = 2 - 3 = -1 \)
2. \( f(f(1)) = f(-1) = 2(-1) - 3 = -2 - 3 = -5 \)

Thus, \( f(f(1)) = -5 \).

---

### 16. \( (f \circ g)(6) \)

This is \( f(g(6)) \). First, we evaluate \( g(6) \), then substitute it into \( f(x) \).

1. \( g(6) = 4 - (6)^2 = 4 - 36 = -32 \)
2. \( f(g(6)) = f(-32) = 2(-32) - 3 = -64 - 3 = -67 \)

Thus, \( (f \circ g)(6) = -67 \).

---

### 17. \( (f \circ f)(5) \)

This is \( f(f(5)) \). First, evaluate \( f(5) \), then substitute it back into \( f(x) \).

1. \( f(5) = 2(5) - 3 = 10 - 3 = 7 \)
2. \( f(f(5)) = f(7) = 2(7) - 3 = 14 - 3 = 11 \)

Thus, \( (f \circ f)(5) = 11 \).

---

### 18. Find the compositions of functions

#### (a) \( f \circ g \)

The composition \( (f \circ g)(x) \) means \( f(g(x)) \). We substitute \( g(x) = 4 - x^2 \) into \( f(x) = 2x - 3 \):

\[
(f \circ g)(x) = f(g(x)) = f(4 - x^2) = 2(4 - x^2) - 3 = 8 - 2x^2 - 3 = 5 - 2x^2
\]

Thus, \( (f \circ g)(x) = 5 - 2x^2 \).

#### (b) \( g \circ f \)

The composition \( (g \circ f)(x) \) means \( g(f(x)) \). We substitute \( f(x) = 2x - 3 \) into \( g(x) = 4 - x^2 \):

\[
(g \circ f)(x) = g(f(x)) = g(2x - 3) = 4 - (2x - 3)^2 = 4 - (4x^2 - 12x + 9) = 4 - 4x^2 + 12x - 9
\]
\[
(g \circ f)(x) = -4x^2 + 12x - 5
\]

Thus, \( (g \circ f)(x) = -4x^2 + 12x - 5 \).

#### (c) \( f \circ f \)

The composition \( (f \circ f)(x) \) is \( f(f(x)) \):

\[
(f \circ f)(x) = f(f(x)) = f(2x - 3) = 2(2x - 3) - 3 = 4x - 6 - 3 = 4x - 9
\]

Thus, \( (f \circ f)(x) = 4x - 9 \).

#### (d) \( g \circ g \)

The composition \( (g \circ g)(x) \) is \( g(g(x)) \):

\[
(g \circ g)(x) = g(g(x)) = g(4 - x^2) = 4 - (4 - x^2)^2 = 4 - (16 - 8x^2 + x^4) = 4 - 16 + 8x^2 - x^4
\]
\[
(g \circ g)(x) = -12 + 8x^2 - x^4
\]

Thus, \( (g \circ g)(x) = -x^4 + 8x^2 - 12 \).

---

### Domains of the functions

- **\( f(x) = 2x - 3 \)**: Domain is all real numbers \( \mathbb{R} \).
- **\( g(x) = 4 - x^2 \)**: Domain is all real numbers \( \mathbb{R} \).
- **\( f \circ g \)**: Domain is all real numbers \( \mathbb{R} \) (since both functions are defined for all \( x \)).
- **\( g \circ f \)**: Domain is all real numbers \( \mathbb{R} \).
- **\( f \circ f \)**: Domain is all real numbers \( \mathbb{R} \).
- **\( g \circ g \)**: Domain is all real numbers \( \mathbb{R} \).

Would you like to go over any part of the solution in more detail?

---

### Related questions:
1. How do you determine the domain of composite functions?
2. What is the difference between \( f \circ g \) and \( g \circ f \)?
3. Can you find the composition of inverse functions?
4. What is the range of \( f(x) = 2x - 3 \)?
5. How do transformations affect the graph of \( f(x) \) and \( g(x) \)?

**Tip**: When composing functions, always evaluate the inner function first and then substitute its result into the outer function.

14. f(g(2))
15. f(f(1))
16. (f ∘ g)(6)
17. (f ∘ f)(5)
Find the functions f ∘ g, g ∘ f, f ∘ f, and g ∘ g and their domains.
18. f(x) = 2x + 3, g(x) = 4x - 1
the given is f(x) = 2x - 3 and g(x) = 4 - x²

Learn how to solve function compositions like f(g(2)) and f(f(1)) using the given functions f(x) = 2x - 3 and g(x) = 4 - x². Detailed step-by-step solutions for function compositions and domain calculations, suitable for students in Grades 10-12.

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