 Direction: Find the inverse of the following functions. Construct a table of values for each function with a limit of -2 to 2, and find its domain and range. with graph
1. 𝑓(𝑥) = 3𝑥 − 2

### Graph Analysis

The graph above shows:
- The blue line representing the function \(f(x) = 3x - 2\).
- The red line representing the inverse function \(f^{-1}(x) = \frac{x + 2}{3}\).
- The green dashed line representing \(y = x\), which is the line of reflection for \(f(x)\) and \(f^{-1}(x)\).

### Summary

1. **Inverse Function**: The inverse of \(f(x) = 3x - 2\) is \(f^{-1}(x) = \frac{x + 2}{3}\).
2. **Table of Values**: The values for both functions were calculated for \(x\) from \(-2\) to \(2\).
3. **Domain and Range**: Both the function and its inverse have a domain and range of all real numbers, \(\mathbb{R}\).
4. **Graph**: Both functions are linear and reflect over the line \(y = x\).

Would you like further details or have any questions about this?

Here are some related questions to explore:
1. How do we find the inverse of a function that is not linear?
2. What happens if a function does not have an inverse?
3. How does the derivative of a function relate to its inverse?
4. Can all functions be inverted? Why or why not?
5. What are the geometric properties of the graph of an inverse function?

**Tip:** To verify if two functions are inverses, check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Explore how to find the inverse of the function f(x) = 3x - 2 with a detailed table of values and graph analysis. Understand the domain, range, and graphical reflection of the function and its inverse. Suitable for high school students.

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