Question

Find the inverse of the given function. f(x)= -1/2SQR x+3, x greater than or equal to -3

Answer 1

ANSWER

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

EXPLANATION

A function will have an inverse if and only if it is a one-to-one function.

The given function is

$f(x) = - \frac{1}{2} \sqrt{x + 3} \: \: where \: \: x \geqslant - 3$

To find the inverse of this function, we let

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

Next, we interchange x and y to get,

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

We now solve for y.

We must clear the fraction by multiplying through with -2 to get;

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

Square both sides of the equation to get:

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

$4x^{2} = y + 3$

Add -3 to both sides

$4 {x}^{2} - 3 = y$

Or

$y = 4 {x}^{2}- 3$

This implies that,

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

This is valid if and only if

$x \geqslant - 3$

Answer 2

Answer:

So, the inverse of function

$f(x) = \frac{-1}{2} \sqrt{x+3}$ is $f^{-1}(x)= 4x^2-3$

Step-by-step explanation:

We need to find the inverse of the given function

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

To find the inverse we replace f(x) with y

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

Now, replacing x with y and y with x

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

Now, we will find the value of y in the above equation

Multiplying both sides by -2

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

Taking square on both sides

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

$4x^2 = y+3$  

Finding value of y

$y = 4x^2-3$

Replacing y with f⁻¹(x)

$f⁻¹(x)= 4x^2-3$

So, the inverse of function

$f(x) = \frac{-1}{2} \sqrt{x+3}$ is $f^{-1}(x)= 4x^2-3$