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.

Question

(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:

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

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 $-\mathrm{\infty }$. As $x$ approaches 0 from the right, $f(x)$ approaches $+\mathrm{\infty }$. 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:

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

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 $(-\mathrm{\infty },2)\cup (2,\mathrm{\infty })$
• Range of $f^{-1}(x)$ is $(-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$

Conclusion

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

• Domain: $(-\mathrm{\infty },2)\cup (2,\mathrm{\infty })$
• Range: $(-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$

The response that matches this is:

domain: $(-\mathrm{\infty },2)\cup (2,\mathrm{\infty })$ range: $(-\mathrm{\infty },0)\cup (0,\mathrm{\infty })$