To find the inverse [tex]\( f^{-1}(x) \)[/tex] of the function [tex]\( f(x)=y=\frac{3x}{8+x} \)[/tex], we need to follow a series of algebraic steps. Let's arrange the given equations in the correct sequence:
1. Start with the given function in terms of [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[ x = \frac{3y}{8 + y} \][/tex]
2. Multiply both sides by [tex]\( 8 + y \)[/tex] to clear the denominator:
[tex]\[ x(8 + y) = 3y \][/tex]
3. Distribute [tex]\( x \)[/tex] on the left side:
[tex]\[ 8x + xy = 3y \][/tex]
4. Move all terms involving [tex]\( y \)[/tex] to one side of the equation:
[tex]\[ 8x = 3y - xy \][/tex]
5. Factor [tex]\( y \)[/tex] on the right side:
[tex]\[ 8x = y(3 - x) \][/tex]
6. Solve for [tex]\( y \)[/tex] by dividing both sides by [tex]\( 3 - x \)[/tex]:
[tex]\[ y = \frac{8x}{3 - x} \][/tex]
Hence, the inverse function is:
[tex]\[ y = f^{-1}(x) = \frac{8x}{3 - x} \][/tex]
So the correct sequence of equations 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]
