Algebra II Function Operation Practice worksheet featuring 12 problems with polynomial functions and domain analysis.

Problem Overview:

We are given several functions and are tasked with performing various operations on them (addition, subtraction, and multiplication). After performing the operations, we need to state the domain of each resulting function. Since all the given functions are polynomial functions, their domains are all real numbers, \( \mathbb{R} \), unless otherwise specified.

The functions are:
- \( f(x) = 9x - 16 \)
- \( g(x) = 11x + 14 \)
- \( h(x) = 12x - 7 \)
- \( j(x) = 4x^2 - 9x - 15 \)
- \( k(x) = 5x^2 - 13x - 8 \)
- \( L(x) = 15x^2 + 11x - 6 \)
- \( m(x) = 8x^2 - 3x + 17 \)

Solution:

#### 1. \( f(x) + g(x) \)
\[
f(x) + g(x) = (9x - 16) + (11x + 14) = 9x + 11x - 16 + 14 = 20x - 2
\]
Domain: \( \mathbb{R} \)

#### 2. \( h(x) - j(x) \)
\[
h(x) - j(x) = (12x - 7) - (4x^2 - 9x - 15) = 12x - 7 - 4x^2 + 9x + 15 = -4x^2 + 21x + 8
\]
Domain: \( \mathbb{R} \)

#### 3. \( g(x) \cdot h(x) \)
\[
g(x) \cdot h(x) = (11x + 14)(12x - 7)
\]
Using the distributive property (FOIL method):
\[
(11x + 14)(12x - 7) = 11x \cdot 12x + 11x \cdot (-7) + 14 \cdot 12x + 14 \cdot (-7)
\]
\[
= 132x^2 - 77x + 168x - 98 = 132x^2 + 91x - 98
\]
Domain: \( \mathbb{R} \)

#### 4. \( f(x) \cdot m(x) \)
\[
f(x) \cdot m(x) = (9x - 16)(8x^2 - 3x + 17)
\]
Using the distributive property:
\[
(9x - 16)(8x^2 - 3x + 17) = 9x \cdot 8x^2 + 9x \cdot (-3x) + 9x \cdot 17 + (-16) \cdot 8x^2 + (-16) \cdot (-3x) + (-16) \cdot 17
\]
\[
= 72x^3 - 27x^2 + 153x - 128x^2 + 48x - 272 = 72x^3 - 155x^2 + 201x - 272
\]
Domain: \( \mathbb{R} \)

#### 5. \( L(x) - k(x) \)
\[
L(x) - k(x) = (15x^2 + 11x - 6) - (5x^2 - 13x - 8) = 15x^2 + 11x - 6 - 5x^2 + 13x + 8 = 10x^2 + 24x + 2
\]
Domain: \( \mathbb{R} \)

#### 6. \( j(x) + L(x) \)
\[
j(x) + L(x) = (4x^2 - 9x - 15) + (15x^2 + 11x - 6) = 4x^2 + 15x^2 - 9x + 11x - 15 - 6 = 19x^2 + 2x - 21
\]
Domain: \( \mathbb{R} \)

#### 7. \( g(x) \cdot j(x) \)
\[
g(x) \cdot j(x) = (11x + 14)(4x^2 - 9x - 15)
\]
Using the distributive property:
\[
(11x + 14)(4x^2 - 9x - 15) = 11x \cdot 4x^2 + 11x \cdot (-9x) + 11x \cdot (-15) + 14 \cdot 4x^2 + 14 \cdot (-9x) + 14 \cdot (-15)
\]
\[
= 44x^3 - 99x^2 - 165x + 56x^2 - 126x - 210 = 44x^3 - 43x^2 - 291x - 210
\]
Domain: \( \mathbb{R} \)

#### 8. \( j(x) - m(x) \)
\[
j(x) - m(x) = (4x^2 - 9x - 15) - (8x^2 - 3x + 17) = 4x^2 - 9x - 15 - 8x^2 + 3x - 17 = -4x^2 - 6x - 32
\]
Domain: \( \mathbb{R} \)

#### 9. \( g(x) \cdot L(x) \)
\[
g(x) \cdot L(x) = (11x + 14)(15x^2 + 11x - 6)
\]
Using the distributive property:
\[
(11x + 14)(15x^2 + 11x - 6) = 11x \cdot 15x^2 + 11x \cdot 11x + 11x \cdot (-6) + 14 \cdot 15x^2 + 14 \cdot 11x + 14 \cdot (-6)
\]
\[
= 165x^3 + 121x^2 - 66x + 210x^2 + 154x - 84 = 165x^3 + 331x^2 + 88x - 84
\]
Domain: \( \mathbb{R} \)

#### 10. \( f(x) \cdot h(x) \)
\[
f(x) \cdot h(x) = (9x - 16)(12x - 7)
\]
Using the distributive property:
\[
(9x - 16)(12x - 7) = 9x \cdot 12x + 9x \cdot (-7) + (-16) \cdot 12x + (-16) \cdot (-7)
\]
\[
= 108x^2 - 63x - 192x + 112 = 108x^2 - 255x + 112
\]
Domain: \( \mathbb{R} \)

#### 11. \( j(x) - k(x) + f(x) \)
\[
j(x) - k(x) + f(x) = (4x^2 - 9x - 15) - (5x^2 - 13x - 8) + (9x - 16)
\]
\[
= 4x^2 - 9x - 15 - 5x^2 + 13x + 8 + 9x - 16 = -x^2 + 13x - 23
\]
Domain: \( \mathbb{R} \)

#### 12. \( m(x) - L(x) - h(x) \)
\[
m(x) - L(x) - h(x) = (8x^2 - 3x + 17) - (15x^2 + 11x - 6) - (12x - 7)
\]
\[
= 8x^2 - 3x + 17 - 15x^2 - 11x + 6 - 12x + 7 = -7x^2 - 26x + 30
\]
Domain: \( \mathbb{R} \)

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 20x - 2 \\
2. & -4x^2 + 21x + 8 \\
3. & 132x^2 + 91x - 98 \\
4. & 72x^3 - 155x^2 + 201x - 272 \\
5. & 10x^2 + 24x + 2 \\
6. & 19x^2 + 2x - 21 \\
7. & 44x^3 - 43x^2 - 291x - 210 \\
8. & -4x^2 - 6x - 32 \\
9. & 165x^3 + 331x^2 + 88x - 84 \\
10. & 108x^2 - 255x + 112 \\
11. & -x^2 + 13x - 23 \\
12. & -7x^2 - 26x + 30 \\
\end{array}
}
\]