# Thread: May I have a question about left and right inverse?

1. ## May I have a question about left and right inverse?

Dear everyone.
I am really sorry.
Could you help me out?

1. Consider the function f : R -> R defined by
f(x) = x^2 + 2 if x > 0 and x + 1 if x <= 0.
Find a left inverse of f.

2. Consider the function f : R -> R defined by
f(x) = x^2 - 2 if x > 0, x - 1 if x <= 0.
Find a right inverse of f.

I know if g(f(x)) = x, it has a left inverse and f(g(x)) = x it has a right inverse.
But, as for solving the questions, I can't adapt to it..

Have a great day!

2. ## Re: May I have a question about left and right inverse?

Originally Posted by yanirose
Dear everyone.
I am really sorry.
Could you help me out?

1. Consider the function f : R -> R defined by
f(x) = x^2 + 2 if x > 0 and x + 1 if x <= 0.
Find a left inverse of f.
So you are looking for a function, g, such that g(f(x))= x. Suppose x>0 then we want g(y)= g(x^2+ 2)= x. Since x> 0, y= x^2+ 2> 2. Swap x and y to write y^2+ 2= x and solve for y: $\displaystyle y= \sqrt{x- 2}$. That is, $\displaystyle g(x)= \sqrt{x- 2}$ with domain x> 2.

Suppose $\displaystyle x\le 0$. Then we want g(y)= g(x+ 1)= x. Write y+ 1= x so y= x- 1. g(x)= x- 1 for x< 1.
(Notice that there is a discontinuity in f at x= 0. From the left, the limit is 1 and from the right, 2. That is why the inverse function is not defined for x between 1 and 2.)

Now you try this one.

2. Consider the function f : R -> R defined by
f(x) = x^2 - 2 if x > 0, x - 1 if x <= 0.
Find a right inverse of f.

I know if g(f(x)) = x, it has a left inverse and f(g(x)) = x it has a right inverse.
But, as for solving the questions, I can't adapt to it..

Have a great day!