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) = 3x - 4 $

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(3x - 4) = 3(3x - 4) = 9x - 12 $

✔ So, $ (f \circ h)(x) = 9x - 12 $

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b) $ (h \circ f)(x) = h(f(x)) = h(3x) = 3(3x) - 4 = 9x - 4 $

✔ So, $ (h \circ f)(x) = 9x - 4 $

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Problem 3:

Given:
- $ f(x) = 2x - 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) = 2x - 1 $

✔ So, $ (f \circ g)(x) = 2x - 1 $

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b) $ (g \circ f)(x) = g(f(x)) = g(2x - 1) = 2x - 1 $

✔ So, $ (g \circ f)(x) = 2x - 1 $

> Note: Since $ g(x) = x $, it's the identity function, so composing it with any function returns that function.

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Problem 4:

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

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

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#### Solution:

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

Now plug into $ f(x) $:
$$
f(2x^2 + 4) = (2x^2 + 4)^2 + 3
$$

Expand:
$$
= (4x^4 + 16x^2 + 16) + 3 = 4x^4 + 16x^2 + 19
$$

✔ So, $ (f \circ g)(x) = 4x^4 + 16x^2 + 19 $

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Problem 5:

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

Which of the following represents $ (f \circ g)(x) $?
A) $ 3x^2 - 6x + 5 $
B) $ 3x^2 - 2x + 5 $
C) $ 3x^2 - 6x $
D) $ 3x^2 + 5 $

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#### Solution:

$ (f \circ g)(x) = f(g(x)) = f(x^2 - 2x) = 3(x^2 - 2x) + 5 = 3x^2 - 6x + 5 $

✔ So, the correct answer is: A) $ 3x^2 - 6x + 5 $

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Problem 6:

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

Which of the following represents $ (h \circ f)(x) $?
A) $ 9x^2 - 1 $
B) $ 3x^2 - 1 $
C) $ 9x - 1 $
D) $ 3x^2 - 3 $

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#### Solution:

$ (h \circ f)(x) = h(f(x)) = h(3x) = (3x)^2 - 1 = 9x^2 - 1 $

✔ So, the correct answer is: A) $ 9x^2 - 1 $

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✔ Final Answers Summary:

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

2. a) $ 9x - 12 $
b) $ 9x - 4 $

3. a) $ 2x - 1 $
b) $ 2x - 1 $

4. $ 4x^4 + 16x^2 + 19 $

5. A) $ 3x^2 - 6x + 5 $

6. A) $ 9x^2 - 1 $

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