To find the inverse of the function $g(x) = -\frac{2}{5}x + 3$, follow these steps:

Step 1: Write the function as $y = g(x)$

$y = -\frac{2}{5}x + 3$

Step 2: Swap $x$ and $y$

This step reflects the idea that the inverse function reverses the roles of the input and output. $x = -\frac{2}{5}y + 3$

Step 3: Solve for $y$

To find the inverse, isolate $y$ on one side of the equation.

1. Subtract 3 from both sides: $x - 3 = -\frac{2}{5}y$

2. Multiply both sides by $-\frac{5}{2}$ to solve for $y$: $y = -\frac{5}{2}(x - 3)$

3. Distribute the $-\frac{5}{2}$: $y = -\frac{5}{2}x + \frac{15}{2}$

Step 4: Express the inverse function

$g^{-1}(x) = -\frac{5}{2}x + \frac{15}{2}$

Summary:

The inverse of the function $g(x) = -\frac{2}{5}x + 3$ is $g^{-1}(x) = -\frac{5}{2}x + \frac{15}{2}$

Would you like more details or have any other questions?

Here are some related questions you might consider:

1. What is the graph of $g^{-1}(x)$ and how does it compare to the graph of $g(x)$?
2. How do you find the inverse of a quadratic function?
3. What is the significance of the inverse function in real-world applications?
4. How do you verify if two functions are inverses of each other?
5. What happens if the original function is not one-to-one? Can it still have an inverse?

Tip: When finding the inverse of a function, always check that the original function is one-to-one (each input has a unique output) to ensure that an inverse exists.