Question

Find f plus g​, f minus g​, ​fg, and StartFraction f Over g EndFraction
. Determine the domain for each function.
f left parenthesis x right parenthesis equals StartRoot x EndRoot​; g left parenthesis x right parenthesis equals x minus 4
Question content area bottom
Part 1
​(f plus g​)(x) ​=
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \( (f + g)(x) \), we need to add the functions \( f(x) \) and \( g(x) \):

1. **Given Functions**:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x - 4 \)

2. **Calculating \( (f + g)(x) \)**:
\[
(f + g)(x) = f(x) + g(x) = \sqrt{x} + (x - 4)
\]
\[
(f + g)(x) = \sqrt{x} + x - 4
\]

### Domain of \( (f + g)(x) \)
To determine the domain of \( f + g \), we consider the individual domains of \( f \) and \( g \):
- The domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \).
- The domain of \( g(x) = x - 4 \) is all real numbers.

The overall domain of \( (f + g)(x) \) is the intersection of the two domains, which is:
\[
\text{Domain of } (f + g)(x) = [0, \infty)
\]

---

### Part 2: Calculate \( (f - g)(x) \)
To find \( (f - g)(x) \):
\[
(f - g)(x) = f(x) - g(x) = \sqrt{x} - (x - 4)
\]
\[
(f - g)(x) = \sqrt{x} - x + 4
\]

### Domain of \( (f - g)(x) \)
The domain remains the same as \( (f + g)(x) \):
\[
\text{Domain of } (f - g)(x) = [0, \infty)
\]

---

### Part 3: Calculate \( (fg)(x) \)
To find \( (fg)(x) \):
\[
(fg)(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x - 4)
\]
\[
(fg)(x) = \sqrt{x}(x - 4)
\]

### Domain of \( (fg)(x) \)
The domain is still based on \( f(x) \):
\[
\text{Domain of } (fg)(x) = [0, \infty)
\]

---

### Part 4: Calculate \( \frac{f}{g}(x) \)
To find \( \frac{f}{g}(x) \):
\[
\frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{x - 4}
\]

### Domain of \( \frac{f}{g}(x) \)
We have constraints from both \( f \) and \( g \):
- The function \( f \) requires \( x \geq 0 \).
- The function \( g \) requires \( g(x) \neq 0 \), so \( x - 4 \neq 0 \) or \( x \neq 4 \).

Thus, the domain of \( \frac{f}{g}(x) \) is:
\[
\text{Domain of } \frac{f}{g}(x) = [0, 4) \cup (4, \infty)
\]

---

### Summary
- \( (f + g)(x) = \sqrt{x} + x - 4 \), Domain: \([0, \infty)\)
- \( (f - g)(x) = \sqrt{x} - x + 4 \), Domain: \([0, \infty)\)
- \( (fg)(x) = \sqrt{x}(x - 4) \), Domain: \([0, \infty)\)
- \( \frac{f}{g}(x) = \frac{\sqrt{x}}{x - 4} \), Domain: \([0, 4) \cup (4, \infty)\)