Composition of Functions Worksheet for 11th Grade | Lesson Planet - Free Printable

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Problem 1: Compositions of Functions

Given:
- \( g(x) = x + 5 \)
- \( h(x) = x^2 \)
- \( f(x) = \sqrt{x} \)

Find:
1. \( (g \circ h)(x) \)
2. \( (h \circ g)(x) \)
3. \( (f \circ h)(x) \)

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Let’s solve these step by step.

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1. Find \( (g \circ h)(x) \)

This means: \( g(h(x)) \)

We know:
- \( h(x) = x^2 \)
- So, plug \( h(x) \) into \( g \):
\( g(h(x)) = g(x^2) = x^2 + 5 \)

✔ Answer: \( (g \circ h)(x) = x^2 + 5 \)

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2. Find \( (h \circ g)(x) \)

This means: \( h(g(x)) \)

We know:
- \( g(x) = x + 5 \)
- So, plug \( g(x) \) into \( h \):
\( h(g(x)) = h(x + 5) = (x + 5)^2 \)

Expand:
\( (x + 5)^2 = x^2 + 10x + 25 \)

✔ Answer: \( (h \circ g)(x) = x^2 + 10x + 25 \)

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3. Find \( (f \circ h)(x) \)

This means: \( f(h(x)) \)

We know:
- \( h(x) = x^2 \)
- \( f(x) = \sqrt{x} \)
- So, \( f(h(x)) = f(x^2) = \sqrt{x^2} \)

Now, \( \sqrt{x^2} = |x| \), because square root gives non-negative result.

✔ Answer: \( (f \circ h)(x) = |x| \)

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Problem 2: Evaluate \( (f \circ g)(4) \)

Given:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x + 5 \)

So:
- \( (f \circ g)(x) = f(g(x)) = f(x + 5) = \sqrt{x + 5} \)

Now evaluate at \( x = 4 \):
- \( (f \circ g)(4) = \sqrt{4 + 5} = \sqrt{9} = 3 \)

✔ Answer: \( (f \circ g)(4) = 3 \)

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Problem 3: Find \( (f \circ g)(x) \) and \( (g \circ f)(x) \)

Given:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x^2 \)

#### a) \( (f \circ g)(x) = f(g(x)) = f(x^2) = \sqrt{x^2} = |x| \)

✔ \( (f \circ g)(x) = |x| \)

#### b) \( (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = (\sqrt{x})^2 = x \)

But note: \( \sqrt{x} \) is only defined for \( x \geq 0 \), so domain is restricted.

✔ \( (g \circ f)(x) = x \), for \( x \geq 0 \)

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Problem 4: Given \( (f \circ g)(x) = \frac{3x + 2}{x - 1} \) and \( g(x) = \sqrt{x} \), find \( f(x) \)

We are told:
- \( (f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = \frac{3x + 2}{x - 1} \)

Let \( u = \sqrt{x} \), so \( x = u^2 \)

Then:
- \( f(u) = \frac{3u^2 + 2}{u^2 - 1} \)

So replacing \( u \) with \( x \):

✔ \( f(x) = \frac{3x^2 + 2}{x^2 - 1} \)

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Final Answers Summary:

1. \( (g \circ h)(x) = x^2 + 5 \)
2. \( (h \circ g)(x) = x^2 + 10x + 25 \)
3. \( (f \circ h)(x) = |x| \)
4. \( (f \circ g)(4) = 3 \)
5. \( (f \circ g)(x) = |x| \), \( (g \circ f)(x) = x \) (for \( x \geq 0 \))
6. \( f(x) = \frac{3x^2 + 2}{x^2 - 1} \)

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