Question

Find f+g, f-g and f/g. Determine the domain for each function.
f(x)=3x-7, g(x)=x-3

1. (f+g)(x)= ___ (simplifying your answer)
What is the domain of f+g?

A. The domain of f+g is {_}. (Use a comma to seperate answers as needed.)
B. The domain of f+g is __. (Type your answer in interval notation.)
C. The domain of f+g is empty set.

(f-g)(x)= ____ (Simplify your answer.)

Answer 1

To find \( f + g \), \( f - g \), and \( \frac{f}{g} \) for the functions given:

1. **Calculating \( f + g \)**:

\[
(f + g)(x) = f(x) + g(x) = (3x - 7) + (x - 3)
\]
Combine like terms:
\[
= 3x - 7 + x - 3 = 4x - 10
\]

So,

\[
(f + g)(x) = 4x - 10
\]


**Domain of \( f + g \)**:

A. The domain of \( f + g \) is **all real numbers** since both \( f(x) \) and \( g(x) \) are linear functions.

B. The domain of \( f + g \) in interval notation is: \((- \infty, \infty)\).

C. The domain of \( f + g \) is **not an empty set**.


2. **Calculating \( f - g \)**:

\[
(f - g)(x) = f(x) - g(x) = (3x - 7) - (x - 3)
\]
Again, combine like terms:
\[
= 3x - 7 - x + 3 = 2x - 4
\]

So,

\[
(f - g)(x) = 2x - 4
\]


**Domain of \( f - g \)**:

A. The domain of \( f - g \) is also **all real numbers** since both \( f(x) \) and \( g(x) \) are linear functions.

B. The domain of \( f - g \) in interval notation is: \((- \infty, \infty)\).

C. The domain of \( f - g \) is **not an empty set**.


3. **Calculating \( \frac{f}{g} \)**:

\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 7}{x - 3}
\]

**Domain of \( \frac{f}{g} \)**:

The function \( \frac{f}{g} \) is undefined when the denominator is zero. Set the denominator equal to zero to find the restriction:
\[
x - 3 = 0 \implies x = 3
\]

Therefore, the domain excludes \( 3 \).

A. The domain of \( \frac{f}{g} \) is all real numbers except \( 3 \): \(\{x | x \neq 3\}\).

B. The domain of \( \frac{f}{g} \) in interval notation is: \((- \infty, 3) \cup (3, \infty)\).

C. The domain of \( \frac{f}{g} \) is **not an empty set**.

In summary:

- \( (f + g)(x) = 4x - 10 \)
- Domain of \( f + g \):
- A: all real numbers
- B: \((- \infty, \infty)\)
- C: Not empty

- \( (f - g)(x) = 2x - 4 \)
- Domain of \( f - g \):
- A: all real numbers
- B: \((- \infty, \infty)\)
- C: Not empty

- \( \left(\frac{f}{g}\right)(x) = \frac{3x - 7}{x - 3} \)
- Domain of \( \frac{f}{g} \):
- A: all real numbers except \( 3 \)
- B: \((- \infty, 3) \cup (3, \infty)\)
- C: Not empty