Question

What is the inverse of this function?​

Answer 1

`f^(-1)(x)=4x^2-3`

Explanation:

The Inverse of a Function

Given a function f(x), the inverse of f, called
`f^(-1)(x)` is a function that satisfies:

`f(f^(-1)(x))=x`

The domain of f becomes the range of its inverse and vice-versa.

The procedure to find the inverse of the function is:

* Write the function as a two-variable equation:

* Solve the equation for x.

* Swap the variables

We are given the function:

`\displaystyle f(x)=-(1)/(2)√(x+3)`

Defined in the interval x ≥ -3

Note f is always negative, thus its range is f(x) ≤ 0

Now we find the inverse. Call y=f(x):

`\displaystyle y=-(1)/(2)√(x+3)`

Multiplying by 2:

`\displaystyle 2y=-√(x+3)`

Squaring both sides:

`\displaystyle 4y^2=x+3`

Subtracting 3:

`\displaystyle 4y^2-3=x`

Swapping variables:

`y=4x^2-3`

Thus:

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

The domain of the inverse is the range of f, thus x is restricted to

x ≤ 0