Question

Find: f^-1(x)=2x/2+3x make sure it is 1-1, if so find the inverse and verify by composition in both directions

Answer 1

The function is one-one can be dtermined by using horizontal line test. If horizontal line on the graph of function intersect the function more than once then such function is not one-one. The graph of function is,

Since horizontal lines intersect the curve of function only once so function is one-one.

Determine the inverse of the function.

`y=(2x)/(2+3x)`

Interchange x with y and y with x and simplify the obtain equation for x.

`\begin{gathered} y=(2x)/(2+3x) \\ y\cdot(2+3x)=2x \\ 2y+3xy=2x \\ 3xy-2x=-2y \\ x(3y-2)=-2y \\ x=-(2y)/(3y-2) \end{gathered}`

Substitute y by x for the inverse of function.

`f(x)=(-2x)/(3x-2)`

So inverse of the function is -2x/(3x - 2).

So functions f and g are inverse of each other if,

`f(g(x))=g(f(x))=x`

Check the obtained inverse function by composition.

`\begin{gathered} f^(-1)(-(2x)/(3x-2))=(2\cdot(-(2x)/(3x-2)))/(2+3\cdot(-(2x)/(3x-2))) \\ =(-(4x)/(3x-2))/((6x-4-6x)/(3x-2)) \\ =-(4x)/(-4) \\ =x \end{gathered}`