Composition of Functions Worksheet | PDF - Free Printable

Let’s solve each part step by step.

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Part I:

Given:
- f(x) = 2x – 1
- g(x) = 3x
- h(x) = x² + 1

We compute compositions by plugging one function into another.

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1. f(g(x))

g(x) = 3x → plug into f:
f(3x) = 2*(3x) – 1 = 6x – 1

✔ Answer: 6x – 1

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2. f(h(x))

h(x) = x² + 1 → plug into f:
f(x² + 1) = 2*(x² + 1) – 1 = 2x² + 2 – 1 = 2x² + 1

✔ Answer: 2x² + 1

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3. g(h(x))

h(x) = x² + 1 → plug into g:
g(x² + 1) = 3*(x² + 1) = 3x² + 3

✔ Answer: 3x² + 3

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4. h(f(x))

f(x) = 2x – 1 → plug into h:
h(2x – 1) = (2x – 1)² + 1
= (4x² – 4x + 1) + 1 = 4x² – 4x + 2

✔ Answer: 4x² – 4x + 2

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5. g(f(x))

f(x) = 2x – 1 → plug into g:
g(2x – 1) = 3*(2x – 1) = 6x – 3

✔ Answer: 6x – 3

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6. h(g(x))

g(x) = 3x → plug into h:
h(3x) = (3x)² + 1 = 9x² + 1

✔ Answer: 9x² + 1

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Part II:

Given:
- f(x) = 9 – x
- g(x) = x² + x
- h(x) = x – 2

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7. g(f(x))

f(x) = 9 – x → plug into g:
g(9 – x) = (9 – x)² + (9 – x)
First, expand (9 – x)² = 81 – 18x + x²
Then add (9 – x):
→ 81 – 18x + x² + 9 – x = x² – 19x + 90

✔ Answer: x² – 19x + 90

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8. f(g(x))

g(x) = x² + x → plug into f:
f(x² + x) = 9 – (x² + x) = 9 – x² – x or –x² – x + 9

✔ Answer: –x² – x + 9

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9. h(f(x))

f(x) = 9 – x → plug into h:
h(9 – x) = (9 – x) – 2 = 7 – x

✔ Answer: 7 – x

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Part III:

m(x) = 10 + 35x → strozzles to vinkles
r(x) = x² + 2 → vinkles to quoons

To go from strozzles directly to quoons, we do r(m(x)):

r(m(x)) = r(10 + 35x) = (10 + 35x)² + 2
Expand: (10 + 35x)² = 100 + 700x + 1225x²
Add 2: → 1225x² + 700x + 102

✔ Composite function: 1225x² + 700x + 102

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Part IV:

t(x) = 28x – 3.7 → toes to moles
z(x) = 3x² – 1 → moles to rolls

Composite: z(t(x)) = z(28x – 3.7) = 3*(28x – 3.7)² – 1

First, compute (28x – 3.7)²:

= (28x)² – 2*28x*3.7 + (3.7)²
= 784x² – 207.2x + 13.69

Now multiply by 3:

3*(784x² – 207.2x + 13.69) = 2352x² – 621.6x + 41.07

Subtract 1:

→ 2352x² – 621.6x + 40.07

✔ Composite function: 2352x² – 621.6x + 40.07

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Final Answer:

I.
1. 6x – 1
2. 2x² + 1
3. 3x² + 3
4. 4x² – 4x + 2
5. 6x – 3
6. 9x² + 1

II.
7. x² – 19x + 90
8. –x² – x + 9
9. 7 – x

III. 1225x² + 700x + 102

IV. 2352x² – 621.6x + 40.07