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 $f^{-1}(f(x)) = x$.