if g(x)=9x+2/x+8 then find range fo rg^-1(x

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 \( \mathbb{R} \setminus \{-8\} \).

Would you like to see a step-by-step breakdown of how the inverse function is calculated or explore related questions?

### 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.

if g(x)=9x+2/x+8 then find range of g^-1(x)

Explore how to find the range of the inverse function g^-1(x) for the rational function g(x) = (9x + 2) / (x + 8). This detailed explanation covers key concepts in algebra, including inverse functions, domains, ranges, and asymptotes. Suitable for high school students in Grades 11-12.

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