Question

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

Answer 1

`\mathrm{Inverse\:of}\:(4x)/(3-x) \text{ is } (3x)/(x+4)`

Solution:

Given that we have to find the inverse function

`g(x) = (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=(4x)/(3-x)`

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

`x=(4y)/(3-y)`

Now solve the above expression for "y"

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

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

`g^(-1)(x) = (3x)/(x+4)`

Thus we have got,

`\mathrm{Inverse\:of}\:(4x)/(3-x) \text{ is } (3x)/(x+4)`