Question

How do you find the inverse of `y=((x^2)-4)/x` and is it a function?

Answer 1

`y=(x+-sqrt(x^2+16))/2`, not a single function

Explanation:

Graph the original function:

graph{(x^2-4)/x [-41.1, 41.1, -20.54, 20.55]}

Since it does not pass the horizontal line test, its inverse will not be a function.

To find its inverse still, flip the `x` terms and `y` terms and then solve for `y`.

`y=(x^2-4)/x" "=>" "x=(y^2-4)/y`

Multiply both sides by `y`.

`xy=y^2-4`

Rearrange to get both terms with `y` on the same side of the equation.

`y^2-xy=4`

Now, add `x^2/4` to both sides of the function.

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

Note that `y^2-xy+x^2/4=(y-x/2)^2` and `x^2/4+4=(x^2+16)/4`.

`(y-x/2)^2=(x^2+16)/4`

Take the square root of both sides. Note that we will take the positive and negative versions of this -- this will actually create two separate functions that cannot act as a function on their own since together they break the vertical line test.

`y-x/2=(+-sqrt(x^2+16))/2`

`y=(x+-sqrt(x^2+16))/2`

Graphed, this should be a reflection of the graph of `y=(x^2+4)/x` over the line `y=x`.

graph{(y-(x+sqrt(x^2+16))/2)(y-(x-sqrt(x^2+16))/2)=0 [-52.02, 52.02, -26, 26.02]}

Note that the top line is the graph of `y=(x+sqrt(x^2+16))/2` and the bottom line is the graph of `y=(x-sqrt(x^2+16))/2`.