[tex]\mathrm{Inverse\:of}\:\frac{4x}{3-x} \text{ is } \frac{3x}{x+4}[/tex]
Solution:
Given that we have to find the inverse function
[tex]g(x) = \frac{4x}{3-x}[/tex]
If a function f(x) is mapping x to y, then the inverse function of f(x) maps y back to x
[tex]y=\frac{4x}{3-x}[/tex]
[tex]\mathrm{Interchange\:the\:variables}\:x\:\mathrm{and}\:y[/tex]
[tex]x=\frac{4y}{3-y}[/tex]
Now solve the above expression for "y"
[tex]x=\frac{4y}{3-y}\\\\\mathrm{Multiply\:both\:sides\:by\:}3-y\\\\x\left(3-y\right)=\frac{4y}{3-y}\left(3-y\right)[/tex]
[tex]x(3-y) = 4y[/tex]
[tex]\mathrm{Expand\:}x\left(3-y\right):\quad 3x-xy[/tex]
[tex]3x-xy = 4y[/tex]
[tex]\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}\\\\3x-xy-3x=4y-3x\\\\\mathrm{Simplify}\\\\-xy=4y-3x\\\\\mathrm{Subtract\:}4y\mathrm{\:from\:both\:sides}\\\\-xy-4y=4y-3x-4y\\\\\mathrm{Simplify}\\\\-xy-4y=-3x\\\\[/tex]
[tex]\mathrm{Factor\:out\:common\:term\:}y\\\\-y(x+4) = -3x[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}-\left(x+4\right)\\\\\frac{-y\left(x+4\right)}{-\left(x+4\right)}=\frac{-3x}{-\left(x+4\right)}\\\\\mathrm{Simplify}\\\\y = \frac{-3x}{-(x+4)}\\\\\text{Cancel the negative sign in numerator and denominator }\\\\y = \frac{3x}{(x+4)}[/tex]
[tex]\text{ Replace y with } g^{-1}(x)[/tex]
[tex]g^{-1}(x) = \frac{3x}{x+4}[/tex]
Thus we have got,
[tex]\mathrm{Inverse\:of}\:\frac{4x}{3-x} \text{ is } \frac{3x}{x+4}[/tex]