Question

Find the inverse function in slope-intercept form (mx+b):
f, of, x, equals, start fraction, 5, divided by, 3, end fraction, x, minus, 10
f(x)=
3
5
​
x−10

Answer 1

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = \frac{5}{3}x - 10 \), we will follow these steps:

1. First, replace \( f(x) \) with \( y \):
\[
y = \frac{5}{3}x - 10
\]

2. Next, solve for \( x \) in terms of \( y \):
\[
y + 10 = \frac{5}{3}x
\]

Multiply both sides by \( \frac{3}{5} \) to isolate \( x \):
\[
x = \frac{3}{5}(y + 10)
\]

3. Now, replace \( y \) with \( x \) to find the inverse function:
\[
f^{-1}(x) = \frac{3}{5}(x + 10)
\]

4. We can distribute \( \frac{3}{5} \):
\[
f^{-1}(x) = \frac{3}{5}x + \frac{3}{5} \cdot 10
\]
\[
f^{-1}(x) = \frac{3}{5}x + 6
\]

Thus, the inverse function in slope-intercept form is:
\[
f^{-1}(x) = \frac{3}{5}x + 6
\]