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Math Help - May I have a question about left and right inverse?

  1. #1
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    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!
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  2. #2
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    Re: May I have a question about left and right inverse?

    Quote Originally Posted by yanirose View Post
    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: y= \sqrt{x- 2}. That is, g(x)= \sqrt{x- 2} with domain x> 2.

    Suppose 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!
    Follow Math Help Forum on Facebook and Google+

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