Composition of Functions Worksheets - Free Printable

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Problem 1:

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

Find:
a) $ (f \circ g)(x) $
b) $ (g \circ f)(x) $

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#### Solution:

a) $ (f \circ g)(x) = f(g(x)) $

We substitute $ g(x) $ into $ f(x) $:
$$
f(g(x)) = f(2x^2 - 5x) = (2x^2 - 5x) + 3 = 2x^2 - 5x + 3
$$

✔ So, $ (f \circ g)(x) = 2x^2 - 5x + 3 $

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b) $ (g \circ f)(x) = g(f(x)) $

We substitute $ f(x) $ into $ g(x) $:
$$
g(f(x)) = g(x + 3) = 2(x + 3)^2 - 5(x + 3)
$$

Now expand:
$$
= 2(x^2 + 6x + 9) - 5x - 15 = 2x^2 + 12x + 18 - 5x - 15 = 2x^2 + 7x + 3
$$

✔ So, $ (g \circ f)(x) = 2x^2 + 7x + 3 $

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Problem 2:

Given:
- $ f(x) = 3x $
- $ h(x) = -4x $

Find:
a) $ (f \circ h)(x) $
b) $ (h \circ f)(x) $

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#### Solution:

a) $ (f \circ h)(x) = f(h(x)) = f(-4x) = 3(-4x) = -12x $

✔ $ (f \circ h)(x) = -12x $

b) $ (h \circ f)(x) = h(f(x)) = h(3x) = -4(3x) = -12x $

✔ $ (h \circ f)(x) = -12x $

> Note: In this case, both compositions give the same result.

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Problem 3:

Given:
- $ f(x) = 3x - 1 $
- $ g(x) = x $

Find:
a) $ (f \circ g)(x) $
b) $ (g \circ f)(x) $

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#### Solution:

a) $ (f \circ g)(x) = f(g(x)) = f(x) = 3x - 1 $

✔ $ (f \circ g)(x) = 3x - 1 $

b) $ (g \circ f)(x) = g(f(x)) = g(3x - 1) = 3x - 1 $

✔ $ (g \circ f)(x) = 3x - 1 $

> Note: Since $ g(x) = x $, it’s the identity function, so composing it doesn’t change anything.

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Problem 4:

Given:
- $ f(x) = x^2 + 5 $
- $ g(x) = 3x - 4 $

Find:
a) $ (f \circ g)(x) $
b) $ (g \circ f)(x) $

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#### Solution:

a) $ (f \circ g)(x) = f(g(x)) = f(3x - 4) = (3x - 4)^2 + 5 $

Expand:
$$
= 9x^2 - 24x + 16 + 5 = 9x^2 - 24x + 21
$$

✔ $ (f \circ g)(x) = 9x^2 - 24x + 21 $

b) $ (g \circ f)(x) = g(f(x)) = g(x^2 + 5) = 3(x^2 + 5) - 4 = 3x^2 + 15 - 4 = 3x^2 + 11 $

✔ $ (g \circ f)(x) = 3x^2 + 11 $

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Problem 5:

Given:
- $ f(x) = x^2 + 1 $
- $ g(x) = 2x^2 + 3 $

Find: $ (f \circ g)(x) $

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#### Solution:

$ (f \circ g)(x) = f(g(x)) = f(2x^2 + 3) = (2x^2 + 3)^2 + 1 $

Expand:
$$
= (4x^4 + 12x^2 + 9) + 1 = 4x^4 + 12x^2 + 10
$$

✔ $ (f \circ g)(x) = 4x^4 + 12x^2 + 10 $

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Problem 6:

Given:
- $ f(x) = x^2 + 1 $
- $ g(x) = x^2 + 3 $

Which of the following represents $ (f \circ g)(x) $?
Options:
A) $ x^4 + 3 $
B) $ x^4 + 3x^2 + 4 $
C) $ x^4 + 3x^2 + 10 $
D) $ x^4 + 6x^2 + 10 $

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#### Solution:

$ (f \circ g)(x) = f(g(x)) = f(x^2 + 3) = (x^2 + 3)^2 + 1 $

$$
= x^4 + 6x^2 + 9 + 1 = x^4 + 6x^2 + 10
$$

✔ Correct answer: D) $ x^4 + 6x^2 + 10 $

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Problem 7:

Given:
- $ f(x) = x $
- $ g(x) = 3x + 4 $

Which of the following represents $ (g \circ f)(x) $?
Options:
A) $ 3x + 4 $
B) $ 3x + 3 $
C) $ x + 4 $
D) $ 3x $

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#### Solution:

$ (g \circ f)(x) = g(f(x)) = g(x) = 3x + 4 $

✔ Correct answer: A) $ 3x + 4 $

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## ✔ Final Answers Summary:

1. a) $ 2x^2 - 5x + 3 $
b) $ 2x^2 + 7x + 3 $

2. a) $ -12x $
b) $ -12x $

3. a) $ 3x - 1 $
b) $ 3x - 1 $

4. a) $ 9x^2 - 24x + 21 $
b) $ 3x^2 + 11 $

5. $ 4x^4 + 12x^2 + 10 $

6. D) $ x^4 + 6x^2 + 10 $

7. A) $ 3x + 4 $

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