Function Operations - Kuta Software - Free Printable

Let's solve each problem step by step from the worksheet titled "Function Operations". We'll perform the indicated operations using the given functions.

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1)

Given:
- $ g(n) = n^2 + 4 + 2n $
- $ h(n) = -3n + 2 $
Find: $ (g - h)(1) $

Step 1: Simplify $ g(n) $
$$
g(n) = n^2 + 2n + 4
$$

Step 2: Compute $ (g - h)(n) = g(n) - h(n) $
$$
(g - h)(n) = (n^2 + 2n + 4) - (-3n + 2) = n^2 + 2n + 4 + 3n - 2 = n^2 + 5n + 2
$$

Step 3: Plug in $ n = 1 $
$$
(1)^2 + 5(1) + 2 = 1 + 5 + 2 = \boxed{8}
$$

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2)

Given:
- $ f(x) = 4x - 3 $
- $ g(x) = x^2 + 2x $
Find: $ (f - g)(4) $

Step 1: Compute $ (f - g)(x) = f(x) - g(x) $
$$
(4x - 3) - (x^2 + 2x) = 4x - 3 - x^2 - 2x = -x^2 + 2x - 3
$$

Step 2: Plug in $ x = 4 $
$$
-(4)^2 + 2(4) - 3 = -16 + 8 - 3 = \boxed{-11}
$$

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3)

Given:
- $ h(x) = 3x + 3 $
- $ g(x) = -4x + 1 $
Find: $ (h + g)(10) $

Step 1: Compute $ (h + g)(x) = h(x) + g(x) $
$$
(3x + 3) + (-4x + 1) = -x + 4
$$

Step 2: Plug in $ x = 10 $
$$
-10 + 4 = \boxed{-6}
$$

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4)

Given:
- $ g(a) = 3a + 2 $
- $ f(a) = 2a - 4 $
Find: $ \left(\frac{g}{f}\right)(3) $

Step 1: Compute $ \frac{g(a)}{f(a)} = \frac{3a + 2}{2a - 4} $

Step 2: Plug in $ a = 3 $
$$
\frac{3(3) + 2}{2(3) - 4} = \frac{9 + 2}{6 - 4} = \frac{11}{2} = \boxed{\frac{11}{2}}
$$

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5)

Given:
- $ g(x) = 2x - 5 $
- $ h(x) = 4x + 5 $
Find: $ g(3) - h(3) $

Step 1: Evaluate $ g(3) $
$$
g(3) = 2(3) - 5 = 6 - 5 = 1
$$

Step 2: Evaluate $ h(3) $
$$
h(3) = 4(3) + 5 = 12 + 5 = 17
$$

Step 3: Subtract
$$
g(3) - h(3) = 1 - 17 = \boxed{-16}
$$

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6)

Given:
- $ g(a) = 2a - 1 $
- $ h(a) = 3a - 3 $
Find: $ (g - h)(-4) $

Step 1: Compute $ (g - h)(a) = (2a - 1) - (3a - 3) = 2a - 1 - 3a + 3 = -a + 2 $

Step 2: Plug in $ a = -4 $
$$
-(-4) + 2 = 4 + 2 = \boxed{6}
$$

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7)

Given:
- $ g(t) = t^2 + 3 $
- $ h(t) = 4t - 3 $
Find: $ (g \cdot h)(-1) $

Step 1: Compute $ (g \cdot h)(t) = g(t) \cdot h(t) $
$$
(t^2 + 3)(4t - 3)
$$

Use distributive property:
$$
= t^2(4t - 3) + 3(4t - 3) = 4t^3 - 3t^2 + 12t - 9
$$

Step 2: Plug in $ t = -1 $
$$
4(-1)^3 - 3(-1)^2 + 12(-1) - 9 = -4 - 3 - 12 - 9 = \boxed{-28}
$$

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8)

Given:
- $ g(n) = 3n + 2 $
- $ f(n) = 2n^2 + 5 $
Find: $ (f \cdot g)(2) $

Step 1: Compute $ (f \cdot g)(n) = f(n) \cdot g(n) $
$$
(2n^2 + 5)(3n + 2)
$$

Multiply:
$$
= 2n^2(3n + 2) + 5(3n + 2) = 6n^3 + 4n^2 + 15n + 10
$$

Step 2: Plug in $ n = 2 $
$$
6(8) + 4(4) + 15(2) + 10 = 48 + 16 + 30 + 10 = \boxed{104}
$$

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9)

Given:
- $ g(x) = -x^2 - 1 - 2x $
- $ f(x) = x + 5 $
Find: $ (g - f)(x) $

Step 1: Simplify $ g(x) $
$$
g(x) = -x^2 - 2x - 1
$$

Step 2: Compute $ (g - f)(x) = g(x) - f(x) $
$$
(-x^2 - 2x - 1) - (x + 5) = -x^2 - 2x - 1 - x - 5 = -x^2 - 3x - 6
$$

Answer: $ \boxed{-x^2 - 3x - 6} $

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10)

Given:
- $ f(x) = 3x - 1 $
- $ g(x) = x^2 - x $
Find: $ \left(\frac{f}{g}\right)(x) $

$$
\left(\frac{f}{g}\right)(x) = \frac{3x - 1}{x^2 - x}
$$

Simplify denominator: $ x^2 - x = x(x - 1) $

So,
$$
\boxed{\frac{3x - 1}{x(x - 1)}}
$$

(Note: This is the expression; no further simplification unless specified.)

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11)

Given:
- $ g(a) = -3a - 3 $
- $ f(a) = a^2 + 5 $
Find: $ (g - f)(a) $

Compute:
$$
(g - f)(a) = g(a) - f(a) = (-3a - 3) - (a^2 + 5) = -3a - 3 - a^2 - 5 = -a^2 - 3a - 8
$$

Answer: $ \boxed{-a^2 - 3a - 8} $

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12)

Given:
- $ h(t) = 2t + 1 $
- $ g(t) = 2t + 2 $
Find: $ (h - g)(t) $

$$
(h - g)(t) = (2t + 1) - (2t + 2) = 2t + 1 - 2t - 2 = \boxed{-1}
$$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ \boxed{8} $ |
| 2 | $ \boxed{-11} $ |
| 3 | $ \boxed{-6} $ |
| 4 | $ \boxed{\frac{11}{2}} $ |
| 5 | $ \boxed{-16} $ |
| 6 | $ \boxed{6} $ |
| 7 | $ \boxed{-28} $ |
| 8 | $ \boxed{104} $ |
| 9 | $ \boxed{-x^2 - 3x - 6} $ |
| 10 | $ \boxed{\frac{3x - 1}{x(x - 1)}} $ |
| 11 | $ \boxed{-a^2 - 3a - 8} $ |
| 12 | $ \boxed{-1} $ |

Let me know if you'd like these checked or explained further!