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pshichka [43]
3 years ago
15

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

Mathematics
2 answers:
tresset_1 [31]3 years ago
5 0

<u>ANSWER</u>

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

<u>EXPLANATION</u>

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

gregori [183]3 years ago
4 0

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

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