Question

find inverse function of:
g(x) = 4x/3-x

Answer 1

$\mathrm{Inverse\:of}\:\frac{4x}{3-x} \text{ is } \frac{3x}{x+4}$

Solution:

Given that we have to find the inverse function

$g(x) = \frac{4x}{3-x}$

If a function f(x) is mapping x to y, then the inverse function of f(x) maps y back to x

$y=\frac{4x}{3-x}$

$\mathrm{Interchange\:the\:variables}\:x\:\mathrm{and}\:y$

$x=\frac{4y}{3-y}$

Now solve the above expression for "y"

$x=\frac{4y}{3-y}\\\\\mathrm{Multiply\:both\:sides\:by\:}3-y\\\\x\left(3-y\right)=\frac{4y}{3-y}\left(3-y\right)$

$x(3-y) = 4y$

$\mathrm{Expand\:}x\left(3-y\right):\quad 3x-xy$

$3x-xy = 4y$

$\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}\\\\3x-xy-3x=4y-3x\\\\\mathrm{Simplify}\\\\-xy=4y-3x\\\\\mathrm{Subtract\:}4y\mathrm{\:from\:both\:sides}\\\\-xy-4y=4y-3x-4y\\\\\mathrm{Simplify}\\\\-xy-4y=-3x$

$\mathrm{Factor\:out\:common\:term\:}y\\\\-y(x+4) = -3x$

$\mathrm{Divide\:both\:sides\:by\:}-\left(x+4\right)\\\\\frac{-y\left(x+4\right)}{-\left(x+4\right)}=\frac{-3x}{-\left(x+4\right)}\\\\\mathrm{Simplify}\\\\y = \frac{-3x}{-(x+4)}\\\\\text{Cancel the negative sign in numerator and denominator }\\\\y = \frac{3x}{(x+4)}$

$\text{ Replace y with } g^{-1}(x)$

$g^{-1}(x) = \frac{3x}{x+4}$

Thus we have got,

$\mathrm{Inverse\:of}\:\frac{4x}{3-x} \text{ is } \frac{3x}{x+4}$