What is the inverse of a function y=xsqrt(3+x^2)

Answer 1

switch x and y and solve for y

x=ysqr(3+y^2)

square both sides, (but that introduces some extraneous solutions)

x^2 = y^2(3+y^2) = 3y^2 + y^4

y^4 +3y^2 = x^2

complete the square

y^4 + 3y^2 + 9/4 = x^2 + 9/4

(y^2+3/2)^2 = x^2+9/4

y^2 = -3/2 + or - sqr(x^2+9/4)

y = + or - sqr[-3/2 + or -sqr(x^2+9/4]

this may be an inverse, but to be an inverse function, there can only be one y value for a given x value, so you'd have to eliminate a square root, either the + or -. as it is, you have 4 different y values for a given x.

the given function y=xsqr(x^2+3) looks like a parabola, similar to y=xsqr(x^2) = y=xx = x^2

if you try to get an inverse it's a parabola rotated 90 degrees, but it's no longer a function

same problem with y=xsqr(x^2+3)