Question

What is the inverse function of `f(x) = 2x + x^(1/2)`?

Answer 1

`color(blue)(y=(4x+1+sqrt(8x+1))/8`

`color(blue)(y=(4x+1-sqrt(8x+1))/8`

Explanation:

Exchange `x` and `y`:

`x=2y+y^(1/2)`

Subtract `2y`:

`x-2y=y^(1/2)`

Square both sides:

`(x-2y)^2=y`

Expand `LHS`

`x^2-4xy+4y^2=y`

`x^2-4xy+4y^2-y=0`

`4y^2-(4x+1)y+x^2=0`

We now solve this for `y`, using the quadratic formula:

`y=(-(-4x-1)+-sqrt((-4x-1)^2-4(4)(x^2)))/(2(4)`

`y=(-(-4x-1)+sqrt((-4x-1)^2-4(4)(x^2)))/(2(4)`

`color(blue)(y=(4x+1+sqrt(8x+1))/8`

`y=(-(-4x-1)-sqrt((-4x-1)^2-4(4)(x^2)))/(2(4)`

`color(blue)(y=(4x+1-sqrt(8x+1))/8`