Function Operations worksheet with ten problems requiring students to perform operations on given functions.

Let’s solve each problem one by one.

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Problem 1:
Given:
- f(n) = -10n - 5
- h(n) = 5n² - 9
Find: (f • h)(n) → This means multiply the two functions: f(n) × h(n)

So:
(-10n - 5)(5n² - 9)

Multiply term by term:

= (-10n)(5n²) + (-10n)(-9) + (-5)(5n²) + (-5)(-9)
= -50n³ + 90n - 25n² + 45

Arrange in descending order of powers:

→ -50n³ - 25n² + 90n + 45

✔ Checked — correct.

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Problem 2:
Given:
- p(z) = z³ - 7z
- f(z) = 6z + 5
Find: (f/p)(z) → This means f(z) divided by p(z)

So:
(6z + 5) / (z³ - 7z)

We can factor denominator if needed, but since numerator doesn’t cancel with denominator, we leave as is.

→ (6z + 5)/(z³ - 7z)

✔ Correct.

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Problem 3:
Given:
- h(s) = 10s - 3
- f(s) = 9s + 4
Find: (h ∘ f)(s) → This means h(f(s)) — plug f(s) into h

So:
h(f(s)) = h(9s + 4) = 10*(9s + 4) - 3
= 90s + 40 - 3
= 90s + 37

✔ Correct.

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Problem 4:
Given:
- h(y) = 4y + 7
- g(y) = y² + 3
Find: (g ∘ h)(y - 10) → First compute g(h(y)), then replace y with (y - 10)? Wait — actually, it says (g ∘ h)(y - 10), which means apply h first to (y - 10), then apply g to that result.

Wait — let’s read carefully: “Find (g ∘ h)(y - 10)”

That means: g(h(y - 10))

Step 1: Compute h(y - 10)
h(y - 10) = 4*(y - 10) + 7 = 4y - 40 + 7 = 4y - 33

Step 2: Plug into g
g(4y - 33) = (4y - 33)² + 3

Expand (4y - 33)²:
= 16y² - 2*4y*33 + 33² = 16y² - 264y + 1089

Add 3:
= 16y² - 264y + 1089 + 3 = 16y² - 264y + 1092

✔ Correct.

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Problem 5:
Given:
- f(d) = -9d - 7
- h(d) = -10d² - 8
Find: (f • h)(d) → Multiply them

(-9d - 7)(-10d² - 8)

Multiply term by term:

= (-9d)(-10d²) + (-9d)(-8) + (-7)(-10d²) + (-7)(-8)
= 90d³ + 72d + 70d² + 56

Arrange:

→ 90d³ + 70d² + 72d + 56

✔ Correct.

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Problem 6:
Given:
- p(k) = k² + 8k
- f(k) = 7k - 9
Find: (f/p)(k) → f(k)/p(k)

So:
(7k - 9)/(k² + 8k)

Can factor denominator: k(k + 8), but no cancellation with numerator.

→ (7k - 9)/(k² + 8k)

✔ Correct.

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Problem 7:
Given:
- h(m) = 2m + 4
- f(m) = -2m - 8
Find: (h ∘ f)(m) → h(f(m))

Plug f(m) into h:

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

✔ Correct.

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Problem 8:
Given:
- h(n) = 8n + 11
- g(n) = n² - 5
Find: (g ∘ h)(n + 10) → g(h(n + 10))

First: h(n + 10) = 8*(n + 10) + 11 = 8n + 80 + 11 = 8n + 91

Then: g(8n + 91) = (8n + 91)² - 5

Expand (8n + 91)²:
= 64n² + 2*8n*91 + 91² = 64n² + 1456n + 8281

Subtract 5:
= 64n² + 1456n + 8281 - 5 = 64n² + 1456n + 8276

✔ Correct.

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Problem 9:
Given:
- f(k) = 9k - 11
- h(k) = 5k² + 10
Find: (f • h)(k) → Multiply

(9k - 11)(5k² + 10)

Multiply:

= 9k*5k² + 9k*10 -11*5k² -11*10
= 45k³ + 90k - 55k² - 110

Arrange:

→ 45k³ - 55k² + 90k - 110

✔ Correct.

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Problem 10:
Given:
- p(s) = s³ - 5s
- f(s) = -3s - 6
Find: (f/p)(s) → f(s)/p(s)

So:
(-3s - 6)/(s³ - 5s)

Factor numerator and denominator if possible.

Numerator: -3(s + 2)
Denominator: s(s² - 5) → cannot simplify further.

→ (-3s - 6)/(s³ - 5s) or factored form: -3(s + 2)/(s(s² - 5))

Either is acceptable, but usually we leave unfactored unless specified.

→ (-3s - 6)/(s³ - 5s)

✔ Correct.

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Final Answers:

1) -50n³ - 25n² + 90n + 45
2) (6z + 5)/(z³ - 7z)
3) 90s + 37
4) 16y² - 264y + 1092
5) 90d³ + 70d² + 72d + 56
6) (7k - 9)/(k² + 8k)
7) -4m - 12
8) 64n² + 1456n + 8276
9) 45k³ - 55k² + 90k - 110
10) (-3s - 6)/(s³ - 5s)