# if F(x)=(Sqrt(2x+1)-2) find the inverse function F(^-1)(X)

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### 3 Answers

I remeber that when I am looking for an inverse I am looking for a function that does the opposite of this function. So first I switch the x and y.

Leaving

`x=sqrt(2y+1)-2`

Then I solve for y by undoing operations to get the y by itself.

`x=sqrt(2y+1)-2`

`+2 +2 `

`x+2=sqrt(2y+1) `

`(x+2)^2=(sqrt(2y+1))^2`

`(x+2)^2=2y+1`

`-1 = -1`

`(x+2)^2-1=2y`

`((x+2)^2-1)/2=y`

`F^-1(x)=((x+2)^2-1)/2`

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We have to solve an equation y=sqrt(2x+1)-2 for x. It is easy:

y+2 = sqrt(2x+1),

(y+2)^2 = (2x+1),

x = (1/2)*[(y+2)^2 - 1].

We must also note that the original y(x) has values from -2 to +infinity, therefore x(y) is defined for y>=-2 only.

`F(x)=sqrt(2x+1)-2`

Let F(x)=y

`y=sqrt(2x+1)-2`

`(y+2)^2=2x+1`

`x=((y+2)^2-1)/2=(y^2+4y+3)/2`

`:.F^(-1)x=(x^2+4x+3)/2`