Finding the Inverse of a Function and its Continuity

[Kate2010]

Homework Statement

f(-1,+1) -> R is given by f(x) = x/(1 - |x|)

Find the inverse of f.

Are f and the inverse of f continuous?

Homework Equations

The Attempt at a Solution

I have shown that f is 1-1.
f((-1,+1)) -> (-$$\infty$$, +$$\infty$$)

Let y = f^-1(x), so f(y) = x

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

If y = 0 x = 0
If y> 0 y =x(1-y), so y = x/(1+x)
If y< 0 y =x(1+y), so y = x/(1-x)

Can I say if y<0 then x <0 and if y >0 then x >0?
Then y=x/(1+|x|)

Would the domain of the inverse be (-$$\infty$$,+$$\infty$$)?

If this is so, f and the inverse of f are continuous as I know f(x) =x, f(x) = 1, f(x) = |x| are continuous and if f and g are continuous then so are |f|, f+g, f-g, and f/g.

 

[mighty2000]

Your solution is correct. The inverse of f is given by f-1(x) = x/(1 + |x|) and the domain of the inverse is (-∞, +∞). Both f and the inverse of f are continuous since they are composed of continuous functions (division, absolute value, and addition/subtraction).