Question

What is the inverse of this function? (Function in photo)

Answer 1

Given:

The function is

`f(x)=-(1)/(2)√(x+3),x\geq -3`

To find:

The inverse of the given function.

Solution:

We have,

`f(x)=-(1)/(2)√(x+3)`

Put f(x)=y.

`y=-(1)/(2)√(x+3)`

Interchange x and y.

`x=-(1)/(2)√(y+3)`

Isolate the variable y.

`-2x=√(y+3)`

`(-2x)^2=y+3`

`4x^2-3=y`

`y=4x^2-3`

Putting
`y=f^(-1)(x)`, we get

`f^(-1)(x)=4x^2-3`

For
`x\geq -3`,

`x+3\geq 0`

`√(x+3)\geq 0`

`-(1)/(2)√(x+3)\leq 0`

`f(x)\leq 0`

It means for function
`f(x)\leq 0` and for inverse function
`x\leq 0` because the range of the function is the domain of inverse function.

Therefore,
`f^(-1)(x)=4x^2-3` for
`x\leq 0`.