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The given functions are:

- \( f(x) = 2 - x^2 \)
- \( g(x) = x^2 + 2x - 80 \)

We need to find:

1. \( (f + g)(x) \)
2. \( (f - g)(x) \)
3. \( (fg)(x) \)
4. \( \frac{f}{g}(x) \)
5. Determine the domain of each function.

### 1. \( (f + g)(x) \):
\[
(f + g)(x) = f(x) + g(x) = (2 - x^2) + (x^2 + 2x - 80)
\]
Simplifying:
\[
(f + g)(x) = 2 - x^2 + x^2 + 2x - 80 = 2x - 78
\]

**Domain of \( f + g \)**: Since there are no restrictions (like division by zero or square roots of negative numbers), the domain is all real numbers: \( (-\infty, \infty) \).

### 2. \( (f - g)(x) \):
\[
(f - g)(x) = f(x) - g(x) = (2 - x^2) - (x^2 + 2x - 80)
\]
Simplifying:
\[
(f - g)(x) = 2 - x^2 - x^2 - 2x + 80 = -2x^2 - 2x + 82
\]

**Domain of \( f - g \)**: Again, no restrictions, so the domain is all real numbers: \( (-\infty, \infty) \).

### 3. \( (fg)(x) \):
\[
(fg)(x) = f(x) \cdot g(x) = (2 - x^2) \cdot (x^2 + 2x - 80)
\]
We will expand this expression:
\[
(fg)(x) = 2(x^2 + 2x - 80) - x^2(x^2 + 2x - 80)
\]
\[
= 2x^2 + 4x - 160 - (x^4 + 2x^3 - 80x^2)
\]
\[
= -x^4 - 2x^3 + 82x^2 + 4x - 160
\]

**Domain of \( fg \)**: Since this is a polynomial, the domain is all real numbers: \( (-\infty, \infty) \).

### 4. \( \frac{f}{g}(x) \):
\[
\frac{f}{g}(x) = \frac{2 - x^2}{x^2 + 2x - 80}
\]
To find the domain, we need to determine when \( g(x) = 0 \). Solve \( x^2 + 2x - 80 = 0 \) using the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-80)}}{2(1)}
\]
\[
x = \frac{-2 \pm \sqrt{4 + 320}}{2} = \frac{-2 \pm \sqrt{324}}{2} = \frac{-2 \pm 18}{2}
\]
Thus:
\[
x = \frac{-2 + 18}{2} = 8 \quad \text{or} \quad x = \frac{-2 - 18}{2} = -10
\]
So \( g(x) = 0 \) when \( x = 8 \) or \( x = -10 \).

**Domain of \( \frac{f}{g} \)**: The function is undefined at \( x = 8 \) and \( x = -10 \), so the domain is:
\[
(-\infty, -10) \cup (-10, 8) \cup (8, \infty)
\]

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Do you want any further details or explanations? Here are 5 related questions:

1. How do you find the domain of a rational function?
2. Can you explain how to expand polynomials in more detail?
3. What is the significance of the quadratic formula in solving \( g(x) = 0 \)?
4. Could you graph these functions to visualize the domain?
5. How do you handle division by zero when determining the domain?

**Tip**: When simplifying functions, always check for restrictions such as division by zero or negative values inside square roots.

For the functions f(x) = 2 - x^2 and g(x) = x^2 + 2x - 80, find f + g, f - g, fg, and f/g. Determine the domain for each function.

Explore how to perform operations on functions and determine their domains for the functions f(x) = 2 - x^2 and g(x) = x^2 + 2x - 80. This detailed solution covers addition, subtraction, multiplication, and division of functions, including domain restrictions and using the quadratic formula to find critical points. Suitable for students in Grades 10-12.

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