Find the inverse function in slope-intercept form (mx+b):
f, of, x, equals, start fraction, 5, divided by, 2, end fraction, x, minus, 10
f(x)=
2
5
​
 x−10

To find the inverse of the function \( f(x) = \frac{5}{2}x - 10 \), we follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = \frac{5}{2}x - 10
\]

### Step 2: Swap \( x \) and \( y \) 
This is done because we are solving for the inverse function.
\[
x = \frac{5}{2}y - 10
\]

### Step 3: Solve for \( y \)
To isolate \( y \), follow these steps:
1. Add 10 to both sides:
   \[
   x + 10 = \frac{5}{2}y
   \]
2. Multiply both sides by \( \frac{2}{5} \) to cancel out the \( \frac{5}{2} \) on the right:
   \[
   y = \frac{2}{5}(x + 10)
   \]

### Step 4: Rewrite the inverse function
Now that \( y \) is isolated, replace \( y \) with \( f^{-1}(x) \):
\[
f^{-1}(x) = \frac{2}{5}(x + 10)
\]

### Final Answer
The inverse function in slope-intercept form is:
\[
f^{-1}(x) = \frac{2}{5}x + 4
\]

Would you like more details or have any questions? 

Here are some related questions to explore:
1. What is the domain and range of the original function?
2. How do you verify that two functions are inverses of each other?
3. Can a function always have an inverse? Why or why not?
4. What happens to the slope when finding the inverse of a linear function?
5. How does shifting the graph of a function affect its inverse?

**Tip:** When finding the inverse, remember to switch \( x \) and \( y \) and then solve for \( y \).

Find the inverse function in slope-intercept form (mx+b): f(x) = (5/2)x - 10

Learn how to find the inverse of the linear function f(x) = (5/2)x - 10, presented in slope-intercept form. Follow a detailed step-by-step solution, covering key algebraic concepts like inverse functions and solving for variables. Ideal for students in Grades 9-11.

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