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To find the inverse of the function \( h(x) = \frac{5}{2}x + 4 \), follow these steps:

### Step 1: Replace \( h(x) \) with \( y \)
\[
y = \frac{5}{2}x + 4
\]

### Step 2: Swap \( x \) and \( y \) to find the inverse
\[
x = \frac{5}{2}y + 4
\]

### Step 3: Solve for \( y \)
1. Subtract 4 from both sides:
   \[
   x - 4 = \frac{5}{2}y
   \]
2. Multiply both sides by \( \frac{2}{5} \) to isolate \( y \):
   \[
   y = \frac{2}{5}(x - 4)
   \]

### Step 4: Replace \( y \) with \( h^{-1}(x) \)
\[
h^{-1}(x) = \frac{2}{5}(x - 4)
\]

So, the inverse function is:
\[
h^{-1}(x) = \frac{2}{5}(x - 4)
\]

Would you like any further details or have any questions?

### Related Questions:
1. What is the domain and range of the original function \( h(x) = \frac{5}{2}x + 4 \)?
2. How do you verify that two functions are inverses of each other?
3. What is the graph of the function \( h(x) = \frac{5}{2}x + 4 \) and its inverse?
4. How does the slope of the original function relate to the slope of its inverse?
5. How do you find the inverse of a quadratic function?

### Tip:
When finding the inverse of a function, always remember to swap \( x \) and \( y \) and then solve for \( y \). This gives you the equation of the inverse function.

What is the inverse of the function h(x) = (5/2)x + 4?

Learn how to find the inverse of the function h(x) = (5/2)x + 4 with this detailed step-by-step guide. This problem explores key algebraic concepts, such as inverse functions and solving linear equations, suitable for students in Grades 9-11.

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