https://file.aiquickdraw.com/mathbot/file/1726678937653aligu5cy.jpg

Let's tackle the problem step by step, focusing first on finding the inverse of each function.

### 7) Function: \( h(x) = 3 - \frac{3}{5}x \)

To find the inverse of this function:
1. Replace \( h(x) \) with \( y \):  
   \[
   y = 3 - \frac{3}{5}x
   \]
   
2. Swap \( x \) and \( y \) to find the inverse:  
   \[
   x = 3 - \frac{3}{5}y
   \]

3. Solve for \( y \):
   \[
   x - 3 = -\frac{3}{5}y
   \]
   Multiply both sides by \( \frac{-5}{3} \) to isolate \( y \):
   \[
   y = \frac{5}{3}(3 - x)
   \]
   
Thus, the inverse function is:
\[
h^{-1}(x) = \frac{5}{3}(3 - x)
\]

### 8) Function: \( g(x) = \frac{-2x - 8}{3} \)

To find the inverse of this function:
1. Replace \( g(x) \) with \( y \):  
   \[
   y = \frac{-2x - 8}{3}
   \]
   
2. Swap \( x \) and \( y \) to find the inverse:  
   \[
   x = \frac{-2y - 8}{3}
   \]

3. Multiply both sides by 3 to eliminate the denominator:
   \[
   3x = -2y - 8
   \]

4. Solve for \( y \):
   \[
   -2y = 3x + 8
   \]
   \[
   y = \frac{-3x - 8}{2}
   \]

Thus, the inverse function is:
\[
g^{-1}(x) = \frac{-3x - 8}{2}
\]

### Graphing the Functions and their Inverses

To graph both the original and inverse functions, you can plot them on the provided grid:
1. For each function, plot several points by substituting values of \( x \).
2. Reflect the graph across the line \( y = x \) to represent the inverse function.
   
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Do you want more detailed steps for graphing these, or any other clarifications?

Here are five related questions to further expand your understanding:
1. How do you determine if a function is invertible just by looking at its equation?
2. What is the graphical interpretation of a function's inverse?
3. How do you check if the inverse you found is correct?
4. What are the domain and range considerations for these functions and their inverses?
5. How would the procedure change if the function included a quadratic term?

**Tip:** Always check if the function passes the horizontal line test to ensure it has an inverse.

Find the inverse of each function. Then graph the function and its inverse. Functions: h(x) = 3 - (3/5)x and g(x) = (-2x - 8) / 3

Learn how to find the inverse of linear functions h(x) = 3 - (3/5)x and g(x) = (-2x - 8)/3. Step-by-step instructions are provided along with a guide to graph the original functions and their inverses.

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