Question

Find the inverse of the given function.

Answer 1

For this case we must find the inverse of the following function:

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

We follow the steps below:

Replace f(x) with y:

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

We exchange the variables:

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

We solve for "y":

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

Multiply by -2 on both sides of the equation:

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

We raise both sides of the equation to the square to eliminate the radical:

$(\sqrt {y + 3}) ^ 2 = (- 2x) ^ 2\\y + 3 = 4x ^ 2$

We subtract 3 from both sides of the equation:

$y = 4x ^ 2-3$

We change y by f ^ {- 1} (x):

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

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

Answer 2

Answer:

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

Step-by-step explanation:

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

To find : Find the inverse of the given function.

Solution : We have given

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

Step 1: take f(x) = y

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

Step 2 : Inter change y and x.

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

Step 3 : Solve for y

Taking square both sides

$x^{2} = \frac{1}{4}(y+3)$.

On multiply both sides by 4.

$4x^{2} = (y+3)$.

On subtraction both sides by 3.

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

Here, $f(x)^{-1}= y$ is inverse of f(x)

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

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