Composition of Functions worksheets - Free Printable

Let's solve each of these composition of functions problems step by step.

---

1)

Given:
- $ f(x) = 4x - 7 $
- $ y(x) = 3x^2 + 1 $
Find: $ f(y(-1)) $

Step 1: Find $ y(-1) $
$$
y(-1) = 3(-1)^2 + 1 = 3(1) + 1 = 3 + 1 = 4
$$

Step 2: Plug into $ f(x) $:
$$
f(y(-1)) = f(4) = 4(4) - 7 = 16 - 7 = \boxed{9}
$$

---

2)

Given:
- $ f(x) = x^2 + 3x $
- $ h(x) = 4x $
Find: $ f(h(-2)) $

Step 1: Find $ h(-2) $
$$
h(-2) = 4(-2) = -8
$$

Step 2: Plug into $ f(x) $:
$$
f(-8) = (-8)^2 + 3(-8) = 64 - 24 = \boxed{40}
$$

---

3)

Given:
- $ g(t) = -2t + 4 $
- $ f(t) = t + 5 $
Find: $ g(f(3)) $

Step 1: Find $ f(3) $
$$
f(3) = 3 + 5 = 8
$$

Step 2: Plug into $ g(t) $:
$$
g(8) = -2(8) + 4 = -16 + 4 = \boxed{-12}
$$

---

4)

Given:
- $ h(x) = x^2 + 2x $
- $ f(x) = x - 7 $
Find: $ h(f(1)) $

Step 1: Find $ f(1) $
$$
f(1) = 1 - 7 = -6
$$

Step 2: Plug into $ h(x) $:
$$
h(-6) = (-6)^2 + 2(-6) = 36 - 12 = \boxed{24}
$$

---

5)

Given:
- $ f(x) = 2x^2 + x $
- $ y(x) = x + 2 $
Find: $ f(y(-5)) $

Step 1: Find $ y(-5) $
$$
y(-5) = -5 + 2 = -3
$$

Step 2: Plug into $ f(x) $:
$$
f(-3) = 2(-3)^2 + (-3) = 2(9) - 3 = 18 - 3 = \boxed{15}
$$

---

6)

Given:
- $ y(x) = x^3 - 5x $
Find: $ y(y(-1)) $

Step 1: Find $ y(-1) $
$$
y(-1) = (-1)^3 - 5(-1) = -1 + 5 = 4
$$

Step 2: Plug into $ y(x) $ again:
$$
y(4) = (4)^3 - 5(4) = 64 - 20 = \boxed{44}
$$

---

7)

Given:
- $ g(x) = 4x^2 - 2 $
- $ y(x) = 2x $
Find: $ g(y(-3)) $

Step 1: Find $ y(-3) $
$$
y(-3) = 2(-3) = -6
$$

Step 2: Plug into $ g(x) $:
$$
g(-6) = 4(-6)^2 - 2 = 4(36) - 2 = 144 - 2 = \boxed{142}
$$

---

8)

Given:
- $ f(t) = t^2 - 5 $
- $ y(t) = t + 9 $
Find: $ f(y(-4)) $

Step 1: Find $ y(-4) $
$$
y(-4) = -4 + 9 = 5
$$

Step 2: Plug into $ f(t) $:
$$
f(5) = (5)^2 - 5 = 25 - 5 = \boxed{20}
$$

---

9)

Given:
- $ f(x) = 5x^3 $
- $ y(x) = -x + 2 $
Find: $ f(y(2)) $

Step 1: Find $ y(2) $
$$
y(2) = -2 + 2 = 0
$$

Step 2: Plug into $ f(x) $:
$$
f(0) = 5(0)^3 = \boxed{0}
$$

---

10)

Given:
- $ y(x) = x^3 - 10 $
- $ h(x) = x - 2 $
Find: $ y(h(2)) $

Step 1: Find $ h(2) $
$$
h(2) = 2 - 2 = 0
$$

Step 2: Plug into $ y(x) $:
$$
y(0) = (0)^3 - 10 = \boxed{-10}
$$

---

✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 9 |
| 2 | 40 |
| 3 | -12 |
| 4 | 24 |
| 5 | 15 |
| 6 | 44 |
| 7 | 142 |
| 8 | 20 |
| 9 | 0 |
| 10 | -10 |

---

🔍 Explanation Summary:

Composition of functions means plugging one function into another. For $ f(g(x)) $, you:
1. Evaluate the inner function first.
2. Use that result as the input to the outer function.

This is done systematically for each problem above.

Let me know if you'd like a visual explanation or practice worksheet!