# The function `f(x)={2x+2}/{3x-7}` is one to one. Find the inverse and check the answer.

lfryerda | High School Teacher | (Level 2) Educator

Posted on

To find the inverse of the function `y={2x+2}/{3x-7}` , we interchange x and y,  and then solve for y.

`x={2y+2}/{3y-7}`    multiply both sides by the denominator of the right side

`x(3y-7)=2y+2`    distribute on left side

`3yx-7x=2y+2`    move y to left x to right

`3yx-2y=7x+2`    factor y on left

`y(3x-2)=7x+2`    divide

`y={7x+2}/{3x-2}`

The inverse function is `f^{-1}(x)={7x+2}/{3x-2}` .

To check, we perform the composition:

`f(f^{-1}(x))={2({7x+2}/{3x-2})+2}/{3({7x+2}/{3x-2})-7}`   get common denominators

`={{2(7x+2)+2(3x-2)}/{3x-2}}/{{3(7x+2)-7(3x-2)}/{3x-2}}`    simplify numerators

`={{14x+4+6x-4}/{3x-2}}/{{21x+6+21x+14}/{3x-2}}`   invert and multiply

`={14x+4+6x-4}/{3x-2} cdot {3x-2}/{21x+6-21x+14}`   simplify

`={20x}/{20}`

`=x`

Since the composition simplifies to x, the inverse function of `f^{-1}(x)={7x+2}/{3x-2}` is correct.