Question

Determine the domain and range for the inverse of f left parenthesis x right parenthesis equals start fraction 1 over 4 x end fraction plus 2.
(1 point)
Responses

domain: left parenthesis negative infinity comma 0 right parenthesis
range: left parenthesis 2 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis range: Image with alt text: left parenthesis 2 comma infinity right parenthesis

domain: left parenthesis negative infinity comma 2 right parenthesis union left parenthesis 2 comma infinity right parenthesis
range: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma 2 right parenthesis union left parenthesis 2 comma infinity right parenthesis range: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis

domain: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
range: left parenthesis negative infinity comma 2 right parenthesis union left parenthesis 2 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis range: Image with alt text: left parenthesis negative infinity comma 2 right parenthesis union left parenthesis 2 comma infinity right parenthesis

domain: left parenthesis 0 comma infinity right parenthesis
range: left parenthesis negative infinity comma 2 right parenthesis

Answer 1

To find the domain and range of the inverse of the function \( f(x) = \frac{1}{4x} + 2 \), we first need to identify the domain and range of the original function.

### Step 1: Determine the domain of \( f(x) \)

The function \( f(x) = \frac{1}{4x} + 2 \) is defined for all \( x \) except \( x = 0 \), since division by zero is undefined. Thus, the domain of \( f(x) \) is:

\[ \text{Domain of } f(x) = (-\infty, 0) \cup (0, \infty) \]

### Step 2: Determine the range of \( f(x) \)

Next, we need to find the range of \( f(x) \). As \( x \) approaches 0 from the left, \( f(x) \) approaches \( -\infty \). As \( x \) approaches 0 from the right, \( f(x) \) approaches \( +\infty \). The function \( f(x) = \frac{1}{4x} + 2 \) has horizontal asymptotes with behavior that shows \( f(x) \) never actually reaches the value 2. Therefore, the range of \( f(x) \) is:

\[ \text{Range of } f(x) = (-\infty, 2) \cup (2, \infty) \]

### Step 3: Determine the domain and range of the inverse function \( f^{-1}(x) \)

The domain of the inverse function \( f^{-1}(x) \) is equal to the range of the original function \( f(x) \), and the range of the inverse function is equal to the domain of the original function:

- **Domain of \( f^{-1}(x) \)** is \( (-\infty, 2) \cup (2, \infty) \)
- **Range of \( f^{-1}(x) \)** is \( (-\infty, 0) \cup (0, \infty) \)

### Conclusion
Thus, the final answer for the domain and range of the inverse function is:

- Domain: \( (-\infty, 2) \cup (2, \infty) \)
- Range: \( (-\infty, 0) \cup (0, \infty) \)

**The response that matches this is:**

**domain:** \( (-\infty, 2) \cup (2, \infty) \)
**range:** \( (-\infty, 0) \cup (0, \infty) \)