Find the inverse function in slope-intercept form (mx+b):

f, of, x, equals, minus, start fraction, 2, divided by, 3, end fraction, x, plus, 10

*f*(*x*)=

−32​*x*+10

To find the inverse of the function \( f(x) = -\frac{2}{3}x + 10 \) in slope-intercept form, follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = -\frac{2}{3}x + 10
\]

### Step 2: Swap \( x \) and \( y \)
\[
x = -\frac{2}{3}y + 10
\]

### Step 3: Solve for \( y \) (the new function will be the inverse)
1. Subtract 10 from both sides:
\[
x - 10 = -\frac{2}{3}y
\]

2. Multiply both sides by the reciprocal of \(-\frac{2}{3}\), which is \(-\frac{3}{2}\):
\[
y = -\frac{3}{2}(x - 10)
\]

3. Distribute \(-\frac{3}{2}\) across the expression:
\[
y = -\frac{3}{2}x + 15
\]

### Step 4: Write the inverse function
\[
f^{-1}(x) = -\frac{3}{2}x + 15
\]

This is the inverse function in slope-intercept form.

Would you like further details or have any questions? Here are some related problems you might consider:

1. Find the inverse of \( f(x) = \frac{4}{5}x - 7 \).
2. Determine the slope and y-intercept of \( f^{-1}(x) = -\frac{3}{2}x + 15 \).
3. Verify if \( f(f^{-1}(x)) = x \) using the original and inverse functions.
4. Solve for \( f^{-1}(x) \) if \( f(x) = -\frac{2}{3}x + c \), where \( c \) is a constant.
5. Graph the original function and its inverse on the same coordinate plane.

**Tip:** The graph of a function and its inverse are always reflections of each other across the line \( y = x \).

Find the inverse function in slope-intercept form (mx+b): f(x) = -2/3x + 10

Learn how to find the inverse of the linear function f(x) = -2/3x + 10 in slope-intercept form. This detailed step-by-step solution covers key algebraic concepts and theorems, suitable for students in Grades 9-11.

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