Question

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

Answer 1

Given:

The function is

$f(x)=-\dfrac{1}{2}\sqrt{x+3},x\geq -3$

To find:

The inverse of the given function.

Solution:

We have,

$f(x)=-\dfrac{1}{2}\sqrt{x+3}$

Put f(x)=y.

$y=-\dfrac{1}{2}\sqrt{x+3}$

Interchange x and y.

$x=-\dfrac{1}{2}\sqrt{y+3}$

Isolate the variable y.

$-2x=\sqrt{y+3}$

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

$4x^2-3=y$

$y=4x^2-3$

Putting $y=f^{-1}(x)$, we get

$f^{-1}(x)=4x^2-3$

For $x\geq -3$,

$x+3\geq 0$

$\sqrt{x+3}\geq 0$

$-\dfrac{1}{2}\sqrt{x+3}\leq 0$

$f(x)\leq 0$

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 $x\leq 0$.