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.

Learn how to find the inverse of the function g(x) = -2/5x + 3 with detailed steps and explanations. This problem covers inverse functions in algebra, including the process of swapping variables, solving equations, and verifying the result. Suitable for students in Grades 10-12.

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