Free function operations worksheet for algebra students to practice combining functions through various mathematical operations.

Let's solve each of the function operations step by step.

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1) $ f(n) = -10n - 5 $, $ h(n) = 5n^2 - 9 $

Find $ (f \circ h)(n) $

This means: $ (f \circ h)(n) = f(h(n)) $

First, substitute $ h(n) $ into $ f(n) $:

$$
f(h(n)) = f(5n^2 - 9) = -10(5n^2 - 9) - 5
$$

Now simplify:

$$
= -50n^2 + 90 - 5 = -50n^2 + 85
$$

✔ Answer: $ -50n^2 + 85 $

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2) $ p(z) = z^2 - 7z $, $ f(z) = 6z + 5 $

Find $ \left(\frac{f}{p}\right)(z) $

This is: $ \left(\frac{f}{p}\right)(z) = \frac{f(z)}{p(z)} = \frac{6z + 5}{z^2 - 7z} $

We can factor denominator:

$$
z^2 - 7z = z(z - 7)
$$

So:

$$
\left(\frac{f}{p}\right)(z) = \frac{6z + 5}{z(z - 7)}
$$

✔ Answer: $ \frac{6z + 5}{z(z - 7)} $

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3) $ h(s) = 10s - 3 $, $ f(s) = 9s + 4 $

Find $ (h \circ f)(s) $

This means: $ h(f(s)) $

Substitute $ f(s) $ into $ h(s) $:

$$
h(f(s)) = h(9s + 4) = 10(9s + 4) - 3 = 90s + 40 - 3 = 90s + 37
$$

✔ Answer: $ 90s + 37 $

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4) $ h(y) = 4y + 7 $, $ g(y) = y^2 + 3 $

Find $ (g \circ h)(y - 10) $

First, find $ (g \circ h)(y) = g(h(y)) $

$$
g(h(y)) = g(4y + 7) = (4y + 7)^2 + 3
$$

Now expand:

$$
(4y + 7)^2 = 16y^2 + 56y + 49
$$

Add 3:

$$
= 16y^2 + 56y + 52
$$

Now evaluate at $ y - 10 $: So replace $ y $ with $ y - 10 $ in the expression above:

$$
g(h(y - 10)) = 16(y - 10)^2 + 56(y - 10) + 52
$$

Compute:

- $ (y - 10)^2 = y^2 - 20y + 100 $
- $ 16(y^2 - 20y + 100) = 16y^2 - 320y + 1600 $
- $ 56(y - 10) = 56y - 560 $
- Add 52

Now sum all:

$$
16y^2 - 320y + 1600 + 56y - 560 + 52 = 16y^2 - 264y + 1092
$$

✔ Answer: $ 16y^2 - 264y + 1092 $

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5) $ f(d) = -9d - 7 $, $ h(d) = -10d^2 - 8 $

Find $ (f \circ h)(d) $

$ f(h(d)) = f(-10d^2 - 8) = -9(-10d^2 - 8) - 7 $

Simplify:

$$
= 90d^2 + 72 - 7 = 90d^2 + 65
$$

✔ Answer: $ 90d^2 + 65 $

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6) $ p(k) = k^2 + 8k $, $ f(k) = 7k - 9 $

Find $ \left(\frac{f}{p}\right)(k) $

$$
\left(\frac{f}{p}\right)(k) = \frac{7k - 9}{k^2 + 8k}
$$

Factor denominator:

$$
k^2 + 8k = k(k + 8)
$$

So:

$$
\frac{7k - 9}{k(k + 8)}
$$

✔ Answer: $ \frac{7k - 9}{k(k + 8)} $

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7) $ h(m) = 2m + 4 $, $ f(m) = -2m - 8 $

Find $ (h \circ f)(m) $

$ h(f(m)) = h(-2m - 8) = 2(-2m - 8) + 4 = -4m - 16 + 4 = -4m - 12 $

✔ Answer: $ -4m - 12 $

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8) $ h(n) = 8n + 11 $, $ g(n) = n^2 - 5 $

Find $ (g \circ h)(n + 10) $

First, $ (g \circ h)(n) = g(h(n)) = g(8n + 11) = (8n + 11)^2 - 5 $

Expand:

$$
(8n + 11)^2 = 64n^2 + 176n + 121
$$

Then subtract 5:

$$
= 64n^2 + 176n + 116
$$

Now evaluate at $ n + 10 $: Replace $ n $ with $ n + 10 $:

$$
g(h(n + 10)) = 64(n + 10)^2 + 176(n + 10) + 116
$$

Compute:

- $ (n + 10)^2 = n^2 + 20n + 100 $
- $ 64(n^2 + 20n + 100) = 64n^2 + 1280n + 6400 $
- $ 176(n + 10) = 176n + 1760 $
- Add 116

Sum all:

$$
64n^2 + 1280n + 6400 + 176n + 1760 + 116 = 64n^2 + 1456n + 8276
$$

✔ Answer: $ 64n^2 + 1456n + 8276 $

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9) $ f(k) = 9k - 11 $, $ h(k) = 5k^2 + 10 $

Find $ (f \circ h)(k) $

$ f(h(k)) = f(5k^2 + 10) = 9(5k^2 + 10) - 11 = 45k^2 + 90 - 11 = 45k^2 + 79 $

✔ Answer: $ 45k^2 + 79 $

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10) $ p(s) = s^3 - 5s $, $ f(s) = -3s - 6 $

Find $ \left(\frac{f}{p}\right)(s) $

$$
\left(\frac{f}{p}\right)(s) = \frac{-3s - 6}{s^3 - 5s}
$$

Factor numerator and denominator:

- Numerator: $ -3(s + 2) $
- Denominator: $ s(s^2 - 5) = s(s - \sqrt{5})(s + \sqrt{5}) $, but we can leave as $ s(s^2 - 5) $

So:

$$
\frac{-3(s + 2)}{s(s^2 - 5)}
$$

✔ Answer: $ \frac{-3(s + 2)}{s(s^2 - 5)} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ -50n^2 + 85 $ |
| 2 | $ \frac{6z + 5}{z(z - 7)} $ |
| 3 | $ 90s + 37 $ |
| 4 | $ 16y^2 - 264y + 1092 $ |
| 5 | $ 90d^2 + 65 $ |
| 6 | $ \frac{7k - 9}{k(k + 8)} $ |
| 7 | $ -4m - 12 $ |
| 8 | $ 64n^2 + 1456n + 8276 $ |
| 9 | $ 45k^2 + 79 $ |
| 10 | $ \frac{-3(s + 2)}{s(s^2 - 5)} $ |

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