Question

Let ​f(x)equalsx squared minus 5 x and ​g(x)equals3 plus x. Find the following.
​(a) left parenthesis f plus g right parenthesis left parenthesis x right parenthesis
​(b) left parenthesis f minus g right parenthesis left parenthesis x right parenthesis
​(c) left parenthesis f times g right parenthesis left parenthesis x right parenthesis
​(d) left parenthesis StartFraction f Over g EndFraction right parenthesis
​(x)
​(e) The domain of StartFraction f Over g EndFraction
Question content area bottom
Part 1
​(a) left parenthesis f plus g right parenthesis left parenthesis x right parenthesisequals
  
enter your response here ​(Simplify your answer. Do not​ factor.)

Answer 1

Let's begin by calculating each part step by step.

Given:
- \( f(x) = x^2 - 5x \)
- \( g(x) = 3 + x \)

### Part (a): \( (f + g)(x) \)

We add \( f(x) \) and \( g(x) \):
\[
(f + g)(x) = f(x) + g(x) = (x^2 - 5x) + (3 + x)
\]
Combining like terms:
\[
= x^2 - 5x + x + 3 = x^2 - 4x + 3
\]
So,
\[
(f + g)(x) = x^2 - 4x + 3.
\]

### Part (b): \( (f - g)(x) \)

Now, we subtract \( g(x) \) from \( f(x) \):
\[
(f - g)(x) = f(x) - g(x) = (x^2 - 5x) - (3 + x)
\]
Combining like terms:
\[
= x^2 - 5x - 3 - x = x^2 - 6x - 3
\]
So,
\[
(f - g)(x) = x^2 - 6x - 3.
\]

### Part (c): \( (f \cdot g)(x) \)

Next, we multiply \( f(x) \) and \( g(x) \):
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x^2 - 5x)(3 + x)
\]
Using the distributive property (FOIL):
\[
= x^2(3) + x^2(x) - 5x(3) - 5x(x) = 3x^2 + x^3 - 15x - 5x^2
\]
Combining like terms:
\[
= x^3 - 2x^2 - 15x
\]
So,
\[
(f \cdot g)(x) = x^3 - 2x^2 - 15x.
\]

### Part (d): \( \left( \frac{f}{g} \right)(x) \)

Now we compute \( \frac{f(x)}{g(x)} \):
\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 5x}{3 + x}
\]
This is the simplified form as is:
\[
\left( \frac{f}{g} \right)(x) = \frac{x^2 - 5x}{3 + x}.
\]

### Part (e): The domain of \( \frac{f}{g} \)

The domain of \( \frac{f}{g}(x) \) is restricted by the denominator:
\[
3 + x \neq 0 \implies x \neq -3
\]
Thus, the domain of \( \frac{f}{g} \) is:
\[
\text{Domain} = \{ x \in \mathbb{R} \,|\, x \neq -3 \}
\]

### Summary of Answers:
(a) \((f + g)(x) = x^2 - 4x + 3\)
(b) \((f - g)(x) = x^2 - 6x - 3\)
(c) \((f \cdot g)(x) = x^3 - 2x^2 - 15x\)
(d) \(\left( \frac{f}{g} \right)(x) = \frac{x^2 - 5x}{3 + x}\)
(e) Domain of \(\left( \frac{f}{g} \right)(x): x \neq -3\)