Composition of Functions Worksheets - Free Printable

It looks like you've uploaded a worksheet titled "Evaluating Composition of Three Functions", and it contains several problems involving function composition. However, since I cannot view or access images directly, I can't see the actual content of the image.

But based on your description and the visible text in your message, I can help you solve the problems if you provide the functions and questions (e.g., what $ f(x) $, $ g(x) $, $ h(x) $ are, and what compositions you're being asked to evaluate).

However, I can already interpret the structure of the worksheet from the text you've provided:

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Problem 1:

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

a) $ f(g(h(2))) $
b) $ h(f(g(1))) $

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Problem 2:

Given:
- $ f(x) = x^2 - 5x^2 $ → Wait, this seems like a typo. Likely meant to be $ f(x) = x^2 - 5x $? Or maybe $ f(x) = x^2 - 5 $? Let's assume it's $ f(x) = x^2 - 5x $.
Wait — actually, $ x^2 - 5x^2 = -4x^2 $. So if that's correct, then $ f(x) = -4x^2 $. But let's double-check.

But looking again:
> "f(x) = x² - 5x²" → That simplifies to $ f(x) = -4x^2 $

Then:
- $ g(x) = \frac{1}{x} $
- $ h(x) = x + 3 $

And we are to evaluate:
a) $ f(g(h(0))) $
b) $ g(f(h(-1))) $

Note: $ g(x) = \frac{1}{x} $, so $ g(0) $ is undefined → likely a trick question.

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Problem 3:

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

Evaluate:
a) $ (f \circ h)(g(6)) $
b) $ (h \circ f)(g(2)) $

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Problem 4:

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

Find:
a) $ (f \circ h)(g(6)) $ → same as problem 3a
b) $ (h \circ f)(g(2)) $ → same as problem 3b

Wait — possibly duplicated?

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Problem 5:

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

Evaluate:
a) $ f(g(h(2))) $
b) $ h(f(g(1))) $

This is similar to problem 1.

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Problem 6:

Let $ f(x) = 2x $, $ g(x) = x + 3 $, $ h(x) = x^2 $

Find:
a) $ f(g(h(2))) $
b) $ h(f(g(1))) $

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Now, let’s go through one full example step-by-step to show how to solve these.

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Example: Solve Problem 1

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

#### a) $ f(g(h(2))) $

We evaluate from the inside out.

1. First, compute $ h(2) $:
$ h(2) = 2^2 = 4 $

2. Next, compute $ g(h(2)) = g(4) $:
$ g(4) = 4 + 5 = 9 $

3. Finally, compute $ f(g(h(2))) = f(9) $:
$ f(9) = 9^2 - 3(9) = 81 - 27 = 54 $

✔ Answer: 54

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#### b) $ h(f(g(1))) $

Again, from inside out.

1. $ g(1) = 1 + 5 = 6 $

2. $ f(g(1)) = f(6) = 6^2 - 3(6) = 36 - 18 = 18 $

3. $ h(f(g(1))) = h(18) = 18^2 = 324 $

✔ Answer: 324

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Now let's do Problem 2, assuming:
- $ f(x) = x^2 - 5x^2 = -4x^2 $ → Important: This may be a typo. But let's proceed.
- $ g(x) = \frac{1}{x} $
- $ h(x) = x + 3 $

#### a) $ f(g(h(0))) $

1. $ h(0) = 0 + 3 = 3 $

2. $ g(h(0)) = g(3) = \frac{1}{3} $

3. $ f(g(h(0))) = f\left(\frac{1}{3}\right) = -4\left(\frac{1}{3}\right)^2 = -4 \cdot \frac{1}{9} = -\frac{4}{9} $

✔ Answer: $-\frac{4}{9}$

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#### b) $ g(f(h(-1))) $

1. $ h(-1) = -1 + 3 = 2 $

2. $ f(h(-1)) = f(2) = -4(2)^2 = -4(4) = -16 $

3. $ g(f(h(-1))) = g(-16) = \frac{1}{-16} = -\frac{1}{16} $

✔ Answer: $-\frac{1}{16}$

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Problem 3:

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

#### a) $ (f \circ h)(g(6)) $

First, $ g(6) = 6 + 1 = 7 $

Then $ (f \circ h)(7) = f(h(7)) $

- $ h(7) = 7 + 3 = 10 $
- $ f(10) = 10^2 = 100 $

✔ Answer: 100

#### b) $ (h \circ f)(g(2)) $

- $ g(2) = 2 + 1 = 3 $
- $ (h \circ f)(3) = h(f(3)) $
- $ f(3) = 3^2 = 9 $
- $ h(9) = 9 + 3 = 12 $

✔ Answer: 12

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Final Notes:

If you can confirm the exact functions (especially for Problem 2 where $ f(x) = x^2 - 5x^2 $ is suspicious), I can adjust accordingly.

But here’s a summary of typical steps:

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🔍 General Strategy for Function Composition:

1. Start from the innermost function and work outward.
2. Evaluate each function step by step.
3. Be careful with domains (e.g., $ \frac{1}{x} $ is undefined at $ x=0 $).
4. Remember:
- $ (f \circ g)(x) = f(g(x)) $
- $ (f \circ g)(a) = f(g(a)) $

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If you can type out the exact functions and questions from your image (or clarify any typos), I can give you full solutions with explanations for all parts.

Would you like me to continue solving the rest?