What is the inverse of y=(1/x)-x?

Question

Determine algebraically if the function is even odd or neither.

Answer 1

Given function y =(1/x)-x

To find inverse of function ,Interchange x and y and solve for y.

x = (1/y)-y

Add y to each side.

x+y = (1/y)-y+y

x+y = 1/y

Multple to each side by y.

y(x+y) = y(1/y)

xy+y^2 = 1

[(y+x/2)^2 = y^2+x^2/4+2*x/2*y = y^2+x^2/4+xy]

Add x^2/4 to each side.

xy+y^2+x^2/4 = 1+x^2/4

(y+x/2)^2 = 1+x^2/4

y+x/2 = ±√(1+x^2/4)

Subtract x/2 from each side.

y+x/2-x/2 = -x/2±√(1+x^2/4)

y = -x/2±√(1+x^2/4)