Question

How do you determine whether the function `h(x)=-4/x^2` has an inverse and if it does, how do you find the inverse function?

Answer 1

See explanation.

Explanation:

If you call it `an` inverse, there is one definition in common use,

symbolized by `h^(-1)x`.

The usually avoided other definition is from x = h^(-1)(y).#

This conforms to the conditions

`h(h^(-1)(y)=y and h^(--1)(h(x))=x`, to mean that

both `h h^(-1) and h^(-1)h` are unit operators, giving the result as the

operand.

In brief, if y is a locally bijective function f(x),

the inverse is f^(-1)(y) = x, giving the same graph, in the

neighborhood.,

I call this inverse `the` inverse.

Here,

`y =-4/x^2 <0`

The inverse is

`x = sqrt(-4/y)`, for `x > 0` and

`x=-sqrt(-4/y)`, for x < 0

I have inserted 1 + 2 = 3 graphs for both,

`y = -4/x^2`, and piecewise inverse.

graph{-4/x^2 [-10, 10, -5, 5]}
graph{x-sqrt(-4/y)=0 [-5, 5 ,-10, 10,]}

graph{x+sqrt(-4/y)=0 [-5, 5 ,-10, 10,]}