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

Question

asked Nov 17, 2022 183k views

1 Answer

AnnieMac · Answer 1 · 2022-11-20T20:17:42+0000

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,
for
$x\leq 0$ .