https://file.aiquickdraw.com/mathbot/file/17272007491415bry11gb.jpg

Let's solve the problem step by step.

### Step 1: Find the inverse function \( g^{-1}(x) \).

Given \( g(x) = \frac{5x}{9x - 5} \), we need to find the inverse function. To do this, we follow these steps:

1. Start by replacing \( g(x) \) with \( y \):

\[
y = \frac{5x}{9x - 5}
\]

2. Swap \( x \) and \( y \) (this is a key step in finding the inverse):

\[
x = \frac{5y}{9y - 5}
\]

3. Solve for \( y \). Multiply both sides by \( 9y - 5 \) to get rid of the denominator:

\[
x(9y - 5) = 5y
\]

4. Expand the left-hand side:

\[
9xy - 5x = 5y
\]

5. Rearrange the equation to group terms involving \( y \) on one side:

\[
9xy - 5y = 5x
\]

6. Factor out \( y \) on the left-hand side:

\[
y(9x - 5) = 5x
\]

7. Solve for \( y \):

\[
y = \frac{5x}{9x - 5}
\]

So, the inverse function is:

\[
g^{-1}(x) = \frac{5x}{9x - 5}
\]

### Step 2: Find the domain and range of the inverse function.

The domain of the inverse function \( g^{-1}(x) \) is the same as the range of the original function \( g(x) \), and the range of the inverse function is the same as the domain of \( g(x) \).

1. **Domain of \( g(x) \):**

   To find the domain of \( g(x) \), we must ensure the denominator does not equal zero:

\[
9x - 5 \neq 0 \implies x \neq \frac{5}{9}
\]

Thus, the domain of \( g(x) \) is:

\[
(-\infty, \frac{5}{9}) \cup (\frac{5}{9}, \infty)
\]

2. **Range of \( g(x) \):**

   The range of \( g(x) \) is all real numbers since \( g(x) \) is a rational function with no horizontal asymptotes restricting the range.

Thus, the range of \( g(x) \) is \( (-\infty, \infty) \).

### Step 3: State the domain and range of \( g^{-1}(x) \).

- The **domain of \( g^{-1}(x) \)** is the range of \( g(x) \), which is \( (-\infty, \infty) \).
- The **range of \( g^{-1}(x) \)** is the domain of \( g(x) \), which is \( (-\infty, \frac{5}{9}) \cup (\frac{5}{9}, \infty) \).

### Final Answer:

\[
g^{-1}(x) = \frac{5x}{9x - 5}
\]

- **Domain of \( g^{-1}(x) \)**: \( (-\infty, \infty) \)
- **Range of \( g^{-1}(x) \)**: \( (-\infty, \frac{5}{9}) \cup (\frac{5}{9}, \infty) \)

Let me know if you'd like any clarifications, or further details on any step.

---

Here are 5 follow-up questions to expand your understanding:

1. How do you verify that two functions are inverses of each other?
2. What role does the domain restriction play in one-to-one functions?
3. Can you find the asymptotes of both \( g(x) \) and \( g^{-1}(x) \)?
4. What happens to the range if we restrict the domain of \( g(x) \)?
5. How does the graph of \( g(x) \) compare to the graph of \( g^{-1}(x) \)?

**Tip:** Always check the function's domain and range before calculating its inverse, as certain values may make the function undefined.

Find g^{-1}(x), where g(x) = \frac{5x}{9x - 5}, and also find the domain and range of g^{-1}(x).

This problem focuses on finding the inverse function g^{-1}(x) for the function g(x) = \frac{5x}{9x - 5}, and determining its domain and range. It covers inverse functions, rational functions, and key steps for solving equations algebraically. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation