The function h is defined by h(x)= x^2-2. for x is less than or equal to 0.Find an expression for h^-1(x)

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lemjay | High School Teacher | (Level 3) Senior Educator

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The given function is:

`h(x)=x^2 -2 `     for `xlt=0` .

To determine its inverse `h^(-1)(x)` , replace h(x) with y.

`y=x^2-2`

Then, switch x and y.

`x=y^2 -2`

And, solve for y. To do so, add both sides by 2.

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

`x+2=y^2`

Take the square root of both sides.

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

`+-sqrt(x+2)=y`

So,

`y=+-sqrt(x+2)`

Then, replace y with `h^(-1)(x)` .

`h^(-1)(x)=+-sqrt(x+2)`

But,  consider the given domain of the the function h(x).

The domain  of h(x) is `xlt=0` . This indicates that the range of the inverse function is `ylt=0` .

Then, `h^(-1)(x)` take only the negative y.

Hence, the inverse function of `h(x)=x^2-2` for `xlt=0` is `h^(-1)(x) = -sqrt(x+2)` .

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