Algebra II worksheet featuring problems on combining functions through various operations.

Let's solve each of the problems step by step. The goal is to perform the indicated operations on the given functions and simplify the results.

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1. Find (f + g)(x)

#### 1. $ f(x) = 5x - 4 $, $ g(x) = 4x - 11 $

$$
(f + g)(x) = f(x) + g(x) = (5x - 4) + (4x - 11) = 9x - 15
$$

✔ Answer: $ 9x - 15 $

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#### 2. $ f(x) = 4x^2 - 2x + 3 $, $ g(x) = -11x^2 + 8x - 1 $

$$
(f + g)(x) = (4x^2 - 2x + 3) + (-11x^2 + 8x - 1) = (4 - 11)x^2 + (-2 + 8)x + (3 - 1) = -7x^2 + 6x + 2
$$

✔ Answer: $ -7x^2 + 6x + 2 $

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#### 3. $ f(x) = 2x^3 - 5x^2 + x - 3 $, $ g(x) = 8x^2 + 5 $

$$
(f + g)(x) = (2x^3 - 5x^2 + x - 3) + (8x^2 + 5) = 2x^3 + (-5 + 8)x^2 + x + (-3 + 5) = 2x^3 + 3x^2 + x + 2
$$

✔ Answer: $ 2x^3 + 3x^2 + x + 2 $

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2. Find (f - g)(x)

#### 4. $ f(x) = 8x - 20 $, $ g(x) = -3x + 19 $

$$
(f - g)(x) = (8x - 20) - (-3x + 19) = 8x - 20 + 3x - 19 = 11x - 39
$$

✔ Answer: $ 11x - 39 $

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#### 5. $ f(x) = x^2 - 9 $, $ g(x) = 7x - 4x + 2 $

Wait: $ g(x) = 7x - 4x + 2 = 3x + 2 $? But it says "7x - 4x + 2", so yes:

$$
g(x) = 3x + 2
$$

Now:
$$
(f - g)(x) = (x^2 - 9) - (3x + 2) = x^2 - 9 - 3x - 2 = x^2 - 3x - 11
$$

✔ Answer: $ x^2 - 3x - 11 $

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#### 6. $ f(x) = 3x^2 + 10x - 4 $, $ g(x) = -6x^2 + x - 8 $

$$
(f - g)(x) = (3x^2 + 10x - 4) - (-6x^2 + x - 8) = 3x^2 + 10x - 4 + 6x^2 - x + 8 = (3+6)x^2 + (10-1)x + (-4+8) = 9x^2 + 9x + 4
$$

✔ Answer: $ 9x^2 + 9x + 4 $

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3. Find (f · g)(x)

#### 7. $ f(x) = x + 3 $, $ g(x) = x - 7 $

$$
(f \cdot g)(x) = (x + 3)(x - 7) = x^2 - 7x + 3x - 21 = x^2 - 4x - 21
$$

✔ Answer: $ x^2 - 4x - 21 $

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#### 8. $ f(x) = 5x^2 $, $ g(x) = 4x^2 + 9x - 3 $

$$
(f \cdot g)(x) = 5x^2(4x^2 + 9x - 3) = 20x^4 + 45x^3 - 15x^2
$$

✔ Answer: $ 20x^4 + 45x^3 - 15x^2 $

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#### 9. $ f(x) = 6x - 1 $, $ g(x) = 2x^2 - 3x + 5 $

$$
(f \cdot g)(x) = (6x - 1)(2x^2 - 3x + 5)
$$

Distribute:

- $ 6x(2x^2) = 12x^3 $
- $ 6x(-3x) = -18x^2 $
- $ 6x(5) = 30x $
- $ -1(2x^2) = -2x^2 $
- $ -1(-3x) = 3x $
- $ -1(5) = -5 $

Add all:

$$
12x^3 + (-18x^2 - 2x^2) + (30x + 3x) - 5 = 12x^3 - 20x^2 + 33x - 5
$$

✔ Answer: $ 12x^3 - 20x^2 + 33x - 5 $

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4. Find $ \left(\frac{f}{g}\right)(x) $

#### 10. $ f(x) = x + 8 $, $ g(x) = x - 2 $

$$
\left(\frac{f}{g}\right)(x) = \frac{x + 8}{x - 2}
$$

✔ Answer: $ \frac{x + 8}{x - 2} $

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#### 11. $ f(x) = 10x^2 - 12x - 8 $, $ g(x) = 2x - 6 $

First factor numerator:

$ 10x^2 - 12x - 8 $

Factor out 2: $ 2(5x^2 - 6x - 4) $

Try factoring: $ 5x^2 - 6x - 4 $

Discriminant: $ 36 + 80 = 116 $ → not perfect square → leave as is.

But let's check if we can simplify:

$$
\frac{10x^2 - 12x - 8}{2x - 6}
$$

Factor denominator: $ 2(x - 3) $

Numerator: $ 10x^2 - 12x - 8 = 2(5x^2 - 6x - 4) $

No common factors, so:

$$
\left(\frac{f}{g}\right)(x) = \frac{10x^2 - 12x - 8}{2x - 6}
$$

We can write as:

$$
= \frac{2(5x^2 - 6x - 4)}{2(x - 3)} = \frac{5x^2 - 6x - 4}{x - 3}, \quad x \neq 3
$$

✔ Answer: $ \frac{5x^2 - 6x - 4}{x - 3} $ (with domain restriction)

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#### 12. $ f(x) = x^2 + 8x + 16 $, $ g(x) = x + 4 $

Note: $ x^2 + 8x + 16 = (x + 4)^2 $

So:

$$
\left(\frac{f}{g}\right)(x) = \frac{(x + 4)^2}{x + 4} = x + 4, \quad x \neq -4
$$

✔ Answer: $ x + 4 $, $ x \neq -4 $

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5. Find $ (f \circ g)(x) $

This means $ f(g(x)) $

#### 13. $ f(x) = 3x $, $ g(x) = 2x + 5 $

$$
f(g(x)) = f(2x + 5) = 3(2x + 5) = 6x + 15
$$

✔ Answer: $ 6x + 15 $

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#### 14. $ f(x) = x + 6 $, $ g(x) = x^2 - 4x + 7 $

$$
f(g(x)) = f(x^2 - 4x + 7) = (x^2 - 4x + 7) + 6 = x^2 - 4x + 13
$$

✔ Answer: $ x^2 - 4x + 13 $

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#### 15. $ f(x) = 4x $, $ g(x) = x^3 + 3x^2 + 2x - 8 $

$$
f(g(x)) = f(x^3 + 3x^2 + 2x - 8) = 4(x^3 + 3x^2 + 2x - 8) = 4x^3 + 12x^2 + 8x - 32
$$

✔ Answer: $ 4x^3 + 12x^2 + 8x - 32 $

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6. Find $ (g \circ f)(x) $

This means $ g(f(x)) $

#### 16. $ f(x) = 3x $, $ g(x) = 2x + 5 $

$$
g(f(x)) = g(3x) = 2(3x) + 5 = 6x + 5
$$

✔ Answer: $ 6x + 5 $

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#### 17. $ f(x) = x + 6 $, $ g(x) = x^2 - 4x + 7 $

$$
g(f(x)) = g(x + 6) = (x + 6)^2 - 4(x + 6) + 7
$$

Compute:

- $ (x + 6)^2 = x^2 + 12x + 36 $
- $ -4(x + 6) = -4x - 24 $
- Add 7

Total:

$$
x^2 + 12x + 36 - 4x - 24 + 7 = x^2 + 8x + 19
$$

✔ Answer: $ x^2 + 8x + 19 $

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#### 18. $ f(x) = 4x $, $ g(x) = x^3 + 3x^2 + 2x - 8 $

$$
g(f(x)) = g(4x) = (4x)^3 + 3(4x)^2 + 2(4x) - 8 = 64x^3 + 3(16x^2) + 8x - 8 = 64x^3 + 48x^2 + 8x - 8
$$

✔ Answer: $ 64x^3 + 48x^2 + 8x - 8 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ 9x - 15 $ |
| 2 | $ -7x^2 + 6x + 2 $ |
| 3 | $ 2x^3 + 3x^2 + x + 2 $ |
| 4 | $ 11x - 39 $ |
| 5 | $ x^2 - 3x - 11 $ |
| 6 | $ 9x^2 + 9x + 4 $ |
| 7 | $ x^2 - 4x - 21 $ |
| 8 | $ 20x^4 + 45x^3 - 15x^2 $ |
| 9 | $ 12x^3 - 20x^2 + 33x - 5 $ |
| 10 | $ \frac{x + 8}{x - 2} $ |
| 11 | $ \frac{5x^2 - 6x - 4}{x - 3} $ |
| 12 | $ x + 4 $, $ x \neq -4 $ |
| 13 | $ 6x + 15 $ |
| 14 | $ x^2 - 4x + 13 $ |
| 15 | $ 4x^3 + 12x^2 + 8x - 32 $ |
| 16 | $ 6x + 5 $ |
| 17 | $ x^2 + 8x + 19 $ |
| 18 | $ 64x^3 + 48x^2 + 8x - 8 $ |

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🔍 Now, about the "quote" puzzle:

The instructions say:
> "Place the letter beside each problem answer blanks below that corresponds to the problem number to find out what the quote is and who said it."

However, in your image, the letters and answer blanks are missing — only the problems are shown. So unless you have a separate key or a list of letters associated with answers, I cannot decode the quote.

But here’s how it usually works:

- Each problem has a number.
- You compute the answer.
- Then, there's a chart or code where each answer maps to a letter.
- You use the problem number to place the letter in order.

Since the mapping isn't provided here, I can't determine the quote or speaker.

But if you provide the letter-to-answer key, I can help decode it.

Alternatively, if this is from a standard worksheet (like from a publisher), the quote might be something like:

> “Mathematics is the language of science.” – Galileo

Or similar.

But without the letter mapping, I can’t proceed further.

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✔ Summary: All math problems solved correctly. If you can provide the letter-key (e.g., "If answer is $ x^2 - 4x - 21 $, then letter is A"), I can reveal the quote.

Let me know!