Question

Find the domain and range of f superscript negative 1 baseline where lower f left parenthesis x right parenthesis equals Start Fraction 1 over 3 x plus 2 End Fraction.
(1 point)
Responses

domain: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
range: left parenthesis negative infinity comma negative Start Fraction 2 over 3 End Fraction right parenthesis union left parenthesis negative Start Fraction 2 over 3 End Fraction 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 negative Start Fraction 2 over 3 End Fraction right parenthesis union left parenthesis negative Start Fraction 2 over 3 End Fraction 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 Start Fraction 2 over 3 End Fraction right parenthesis union left parenthesis Start Fraction 2 over 3 End Fraction 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 Start Fraction 2 over 3 End Fraction right parenthesis union left parenthesis Start Fraction 2 over 3 End Fraction 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 0 right parenthesis union left parenthesis 0 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 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis

domain: left parenthesis negative infinity comma negative Start Fraction 2 over 3 End Fraction right parenthesis union left parenthesis negative Start Fraction 2 over 3 End Fraction comma infinity right parenthesis
range: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis

Answer 1

To find the domain and range of the inverse function \( f^{-1}(x) \) given the function \( f(x) = \frac{1}{3x + 2} \), we can follow these steps:

### Step 1: Find the Domain of \( f(x) \)

The domain of \( f(x) \) is determined by the values of \( x \) that keep the denominator from being zero.

Set the denominator equal to zero to find restrictions:

\[
3x + 2 = 0
\]

Solving for \( x \):

\[
3x = -2 \implies x = -\frac{2}{3}
\]

Thus, the function \( f(x) \) is undefined at \( x = -\frac{2}{3} \). Therefore, the domain of \( f(x) \) is:

\[
\text{Domain of } f: \left( -\infty, -\frac{2}{3} \right) \cup \left( -\frac{2}{3}, \infty \right)
\]

### Step 2: Find the Range of \( f(x) \)

Next, let's determine the range of \( f(x) \).

The function \( f(x) = \frac{1}{3x + 2} \) approaches 0 as \( x \to -\infty \) or \( x \to \infty \), but it never reaches 0. The output can be any value except 0. Therefore, the range of \( f(x) \) is:

\[
\text{Range of } f: \left( -\infty, 0 \right) \cup \left( 0, \infty \right)
\]

### Step 3: Find the Domain and Range of the Inverse Function

Using the properties of inverse functions, we have:

- The domain of \( f^{-1}(x) \) is equal to the range of \( f(x) \).
- The range of \( f^{-1}(x) \) is equal to the domain of \( f(x) \).

Thus, we find:

- The domain of \( f^{-1}(x) \) is \( \left( -\infty, 0 \right) \cup \left( 0, \infty \right) \)
- The range of \( f^{-1}(x) \) is \( \left( -\infty, -\frac{2}{3} \right) \cup \left( -\frac{2}{3}, \infty \right) \)

### Conclusion

Putting it all together, the correct response would be:

**Domain of \( f^{-1}(x) \):** \( \left( -\infty, 0 \right) \cup \left( 0, \infty \right) \)
**Range of \( f^{-1}(x) \):** \( \left( -\infty, -\frac{2}{3} \right) \cup \left( -\frac{2}{3}, \infty \right) \)

None of the provided options exactly match, as they list different domain and range sets. Please verify the options or details provided, or clarify if there was a misunderstanding.