To find the range of the inverse function $g^{-1}(x)$, we need to understand the original function $g(x) = \frac{9x + 2}{x + 8}$.

Step 1: Find the Domain of $g(x)$

The function $g(x)$ is a rational function. The domain of $g(x)$ excludes any values of $x$ that make the denominator zero. Therefore, we solve for when the denominator is zero: $x + 8 = 0 \implies x = -8$ So, the domain of $g(x)$ is $x \in \mathbb{R} \setminus \{-8\}$.

Step 2: Find the Range of $g(x)$

To find the range of $g(x)$, we analyze the function by solving for $y$ in terms of $x$: $y = \frac{9x + 2}{x + 8}$ Multiply both sides by $x + 8$: $y(x + 8) = 9x + 2$ Expand and rearrange the equation: $yx + 8y = 9x + 2$ Move terms involving $x$ to one side: $yx - 9x = 2 - 8y$ Factor out $x$: $x(y - 9) = 2 - 8y$ Solve for $x$: $x = \frac{2 - 8y}{y - 9}$ The function is undefined when $y = 9$. This means that $y = 9$ is a horizontal asymptote, and thus, the range of $g(x)$ is all real numbers except 9: $\text{Range of } g(x) = \mathbb{R} \setminus \{9\}$

Step 3: Find the Range of $g^{-1}(x)$

The range of the inverse function $g^{-1}(x)$ is the same as the domain of the original function $g(x)$. Therefore, the range of $g^{-1}(x)$ is all real numbers except $-8$: $\text{Range of } g^{-1}(x) = \mathbb{R} \setminus \{-8\}$

Final Answer:

The range of $g^{-1}(x)$ is $\mathbb{R} \setminus \{-8\}$.

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Related Questions:

1. How do you find the inverse of a rational function?
2. What are the properties of rational functions?
3. How do horizontal asymptotes affect the range of a function?
4. What is the significance of domain and range in a function?
5. How can you graph a rational function to determine its range?

Tip:

When dealing with rational functions, always check for vertical and horizontal asymptotes as they can significantly affect the domain and range of the function.