Solve x for Inverse of y=sqrt(x^3+x^2+x+1) - Help Needed

[yoyo]

what is the inverse of of y=sqrt(x^3+x^2+x+1)

i know u are suppose to solve for x but having trouble...help please

 

[dextercioby]

Why don't u do it?
x^{3}+x^{2}+x+1=y^{2}

U need to solve this cubic for "x".Use Cardano's formulae.

Daniel.

 

[thepatientmental]

To find the inverse of a function, you need to switch the roles of x and y and solve for y. In this case, we have:

y = sqrt(x^3 + x^2 + x + 1)

Switching the roles of x and y, we get:

x = sqrt(y^3 + y^2 + y + 1)

Now, we need to isolate y on one side of the equation. To do this, we will square both sides:

x^2 = y^3 + y^2 + y + 1

Next, we will rearrange the terms to have the y terms on one side and the constant terms on the other side:

y^3 + y^2 + y = x^2 - 1

Now, we can factor out a y from the left side:

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

Next, we can use the quadratic formula to solve for y^2 + y + 1:

y^2 + y + 1 = (-1 ± sqrt(1^2 - 4(1)(x^2 - 1))) / 2(1)

= (-1 ± sqrt(4x^2 - 3)) / 2

Therefore, our inverse function is:

y = (-1 ± sqrt(4x^2 - 3)) / 2

Note: This is a piecewise function, meaning it has two different branches depending on the value of x. If x is positive, then the inverse function is:

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

If x is negative, then the inverse function is:

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

I hope this helps! If you are still having trouble, I would recommend practicing more with solving equations and using the quadratic formula. Good luck!