Multiplying and Dividing Functions worksheet with 16 problems requiring students to perform function operations.

Problem Overview:

The task involves performing operations on functions, specifically multiplication and division of functions. We are given pairs of functions \( h(x) \) and \( g(x) \), and we need to compute either the product \( (h \cdot g)(x) \) or the quotient \( \left( \frac{g}{h} \right)(x) \). Some problems also require evaluating these results at specific values of \( x \).

General Approach:

1. Multiplication of Functions:
\[
(h \cdot g)(x) = h(x) \cdot g(x)
\]
Multiply the expressions for \( h(x) \) and \( g(x) \).

2. Division of Functions:
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)}
\]
Divide the expression for \( g(x) \) by the expression for \( h(x) \).

3. Evaluation:
If a specific value of \( x \) is given (e.g., \( x = 4 \)), substitute that value into the resulting function after performing the operation.

Solutions:

#### 1. \( h(x) = 9x \), \( g(x) = 4x + 2 \), Find \( (h \cdot g)(4) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (9x) \cdot (4x + 2) = 36x^2 + 18x
\]

- Step 2: Evaluate at \( x = 4 \):
\[
(h \cdot g)(4) = 36(4)^2 + 18(4) = 36 \cdot 16 + 72 = 576 + 72 = 648
\]

- Answer:
\[
\boxed{648}
\]

#### 2. \( h(x) = 2x \), \( g(x) = 8x^3 + 6x^2 \), Find \( \left( \frac{g}{h} \right)(x) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{8x^3 + 6x^2}{2x}
\]
Simplify the expression:
\[
\frac{8x^3 + 6x^2}{2x} = \frac{8x^3}{2x} + \frac{6x^2}{2x} = 4x^2 + 3x
\]

- Answer:
\[
\boxed{4x^2 + 3x}
\]

#### 3. \( h(x) = 10x \), \( g(x) = 30x^3 + 40x^2 \), Find \( \left( \frac{g}{h} \right)(1) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{30x^3 + 40x^2}{10x}
\]
Simplify the expression:
\[
\frac{30x^3 + 40x^2}{10x} = \frac{30x^3}{10x} + \frac{40x^2}{10x} = 3x^2 + 4x
\]

- Step 2: Evaluate at \( x = 1 \):
\[
\left( \frac{g}{h} \right)(1) = 3(1)^2 + 4(1) = 3 + 4 = 7
\]

- Answer:
\[
\boxed{7}
\]

#### 4. \( h(x) = -4x \), \( g(x) = 2x - 3 \), Find \( (h \cdot g)(x) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (-4x) \cdot (2x - 3) = -8x^2 + 12x
\]

- Answer:
\[
\boxed{-8x^2 + 12x}
\]

#### 5. \( h(x) = 7x \), \( g(x) = 8x + 3 \), Find \( (h \cdot g)(1) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (7x) \cdot (8x + 3) = 56x^2 + 21x
\]

- Step 2: Evaluate at \( x = 1 \):
\[
(h \cdot g)(1) = 56(1)^2 + 21(1) = 56 + 21 = 77
\]

- Answer:
\[
\boxed{77}
\]

#### 6. \( h(x) = 3x \), \( g(x) = 6x + 4 \), Find \( (h \cdot g)(3) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (3x) \cdot (6x + 4) = 18x^2 + 12x
\]

- Step 2: Evaluate at \( x = 3 \):
\[
(h \cdot g)(3) = 18(3)^2 + 12(3) = 18 \cdot 9 + 36 = 162 + 36 = 198
\]

- Answer:
\[
\boxed{198}
\]

#### 7. \( h(x) = -2x \), \( g(x) = 4x - 3 \), Find \( (h \cdot g)(x) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (-2x) \cdot (4x - 3) = -8x^2 + 6x
\]

- Answer:
\[
\boxed{-8x^2 + 6x}
\]

#### 8. \( h(x) = 5x \), \( g(x) = -10x^3 + 10x^2 \), Find \( \left( \frac{g}{h} \right)(x) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{-10x^3 + 10x^2}{5x}
\]
Simplify the expression:
\[
\frac{-10x^3 + 10x^2}{5x} = \frac{-10x^3}{5x} + \frac{10x^2}{5x} = -2x^2 + 2x
\]

- Answer:
\[
\boxed{-2x^2 + 2x}
\]

#### 9. \( h(x) = 10x \), \( g(x) = 8x + 3 \), Find \( (h \cdot g)(x) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (10x) \cdot (8x + 3) = 80x^2 + 30x
\]

- Answer:
\[
\boxed{80x^2 + 30x}
\]

#### 10. \( h(x) = 9x \), \( g(x) = -27x^3 + 36x^2 \), Find \( \left( \frac{g}{h} \right)(x) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{-27x^3 + 36x^2}{9x}
\]
Simplify the expression:
\[
\frac{-27x^3 + 36x^2}{9x} = \frac{-27x^3}{9x} + \frac{36x^2}{9x} = -3x^2 + 4x
\]

- Answer:
\[
\boxed{-3x^2 + 4x}
\]

#### 11. \( h(x) = 2x \), \( g(x) = -4x^3 + 8x^2 \), Find \( \left( \frac{g}{h} \right)(3) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{-4x^3 + 8x^2}{2x}
\]
Simplify the expression:
\[
\frac{-4x^3 + 8x^2}{2x} = \frac{-4x^3}{2x} + \frac{8x^2}{2x} = -2x^2 + 4x
\]

- Step 2: Evaluate at \( x = 3 \):
\[
\left( \frac{g}{h} \right)(3) = -2(3)^2 + 4(3) = -2 \cdot 9 + 12 = -18 + 12 = -6
\]

- Answer:
\[
\boxed{-6}
\]

#### 12. \( h(x) = -5x \), \( g(x) = 6x - 2 \), Find \( (h \cdot g)(x) \)

- Step 1: Compute \( (h \cdot g)(x) \):
\[
(h \cdot g)(x) = h(x) \cdot g(x) = (-5x) \cdot (6x - 2) = -30x^2 + 10x
\]

- Answer:
\[
\boxed{-30x^2 + 10x}
\]

#### 13. \( h(x) = 4x \), \( g(x) = 16x^3 + 16x^2 \), Find \( \left( \frac{g}{h} \right)(3) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{16x^3 + 16x^2}{4x}
\]
Simplify the expression:
\[
\frac{16x^3 + 16x^2}{4x} = \frac{16x^3}{4x} + \frac{16x^2}{4x} = 4x^2 + 4x
\]

- Step 2: Evaluate at \( x = 3 \):
\[
\left( \frac{g}{h} \right)(3) = 4(3)^2 + 4(3) = 4 \cdot 9 + 12 = 36 + 12 = 48
\]

- Answer:
\[
\boxed{48}
\]

#### 14. \( h(x) = 6x \), \( g(x) = -18x^3 + 12x^2 \), Find \( \left( \frac{g}{h} \right)(-4) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{-18x^3 + 12x^2}{6x}
\]
Simplify the expression:
\[
\frac{-18x^3 + 12x^2}{6x} = \frac{-18x^3}{6x} + \frac{12x^2}{6x} = -3x^2 + 2x
\]

- Step 2: Evaluate at \( x = -4 \):
\[
\left( \frac{g}{h} \right)(-4) = -3(-4)^2 + 2(-4) = -3 \cdot 16 + (-8) = -48 - 8 = -56
\]

- Answer:
\[
\boxed{-56}
\]

#### 15. \( h(x) = 10x \), \( g(x) = 20x^3 + 20x^2 \), Find \( \left( \frac{g}{h} \right)(x) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{20x^3 + 20x^2}{10x}
\]
Simplify the expression:
\[
\frac{20x^3 + 20x^2}{10x} = \frac{20x^3}{10x} + \frac{20x^2}{10x} = 2x^2 + 2x
\]

- Answer:
\[
\boxed{2x^2 + 2x}
\]

#### 16. \( h(x) = 8x \), \( g(x) = -24x^3 + 32x^2 \), Find \( \left( \frac{g}{h} \right)(2) \)

- Step 1: Compute \( \left( \frac{g}{h} \right)(x) \):
\[
\left( \frac{g}{h} \right)(x) = \frac{g(x)}{h(x)} = \frac{-24x^3 + 32x^2}{8x}
\]
Simplify the expression:
\[
\frac{-24x^3 + 32x^2}{8x} = \frac{-24x^3}{8x} + \frac{32x^2}{8x} = -3x^2 + 4x
\]

- Step 2: Evaluate at \( x = 2 \):
\[
\left( \frac{g}{h} \right)(2) = -3(2)^2 + 4(2) = -3 \cdot 4 + 8 = -12 + 8 = -4
\]

- Answer:
\[
\boxed{-4}
\]

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 648 \\
2. & \ 4x^2 + 3x \\
3. & \ 7 \\
4. & \ -8x^2 + 12x \\
5. & \ 77 \\
6. & \ 198 \\
7. & \ -8x^2 + 6x \\
8. & \ -2x^2 + 2x \\
9. & \ 80x^2 + 30x \\
10. & \ -3x^2 + 4x \\
11. & \ -6 \\
12. & \ -30x^2 + 10x \\
13. & \ 48 \\
14. & \ -56 \\
15. & \ 2x^2 + 2x \\
16. & \ -4 \\
\end{aligned}
}
\]