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?

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Answer 1 · 2017-01-27T05:51:30

See 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,]}