To find the inverse of the function [tex]\( f(x) = y = \frac{3x}{8 + x} \)[/tex], we need to follow these steps. Here's the step-by-step solution with the equations arranged in their correct sequence:
1. Start with the function and express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = \frac{3x}{8 + x}
\][/tex]
2. Rewrite the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3y}{8 + y}
\][/tex]
3. Multiply both sides by [tex]\( 8 + y \)[/tex] to eliminate the fraction:
[tex]\[
x(8 + y) = 3y
\][/tex]
4. Distribute [tex]\( x \)[/tex] on the left-hand side:
[tex]\[
8x + xy = 3y
\][/tex]
5. Get all terms involving [tex]\( y \)[/tex] on one side of the equation:
[tex]\[
8x + xy - 3y = 0
\][/tex]
6. Factor out [tex]\( y \)[/tex]:
[tex]\[
8x = 3y - xy
\][/tex]
7. Combine terms with [tex]\( y \)[/tex]:
[tex]\[
8x = y(3 - x)
\][/tex]
8. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{8x}{3 - x}
\][/tex]
Thus the inverse function is:
[tex]\[
f^{-1}(x) = \frac{8x}{3 - x}
\][/tex]
The correct sequence of equations to find the inverse function is:
1. [tex]\( x = \frac{3y}{8 + y} \)[/tex]
2. [tex]\( x(8 + y) = 3y \)[/tex]
3. [tex]\( 8x + xy = 3y \)[/tex]
4. [tex]\( 8x = 3y - xy \)[/tex]
5. [tex]\( 8x = y(3 - x) \)[/tex]
6. [tex]\( y = f^{-1}(x) = \frac{8x}{3 - x} \)[/tex]
Arranged in this order, the equations correctly lead to the inverse function [tex]\( f^{-1}(x) = \frac{8x}{3 - x} \)[/tex].
