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Math Help - domain of a composite function

  1. #1
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    domain of a composite function

    Say i am given 2 functions , f(x) and g(x) with their respective domains , is it possible for me to know the domain of f o g without evaluating function f o g ?

    Example :

    f(x)=\sqrt{x-1} , x\geq 1

    g(x)=\frac{1}{x-1} , x\in R , x\neq 1


    fg(x)=\sqrt{\frac{1}{x-1}-1}

    i see that the expression under the square roots must be >=0 , so

    1\leq x \leq 2 but since x\neq 1 so the domain is 1<x\leq 2

    Thats how i would do it . Is there an easier way ?
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  2. #2
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    Quote Originally Posted by hooke View Post
    Say i am given 2 functions , f(x) and g(x) with their respective domains , is it possible for me to know the domain of f o g without evaluating function f o g ?

    Example :

    f(x)=\sqrt{x-1} , x\geq 1

    g(x)=\frac{1}{x-1} , x\in R , x\neq 1


    fg(x)=\sqrt{\frac{1}{x-1}-1}

    i see that the expression under the square roots must be >=0 , so

    1\leq x \leq 2 but since x\neq 1 so the domain is 1<x\leq 2

    Thats how i would do it . Is there an easier way ?
    x will be in the domain of fog if it is in the domain of g and g(x) is in the domain of f.

    Here, g(x)= \frac{1}{x- 1} so its domain is all real numbers except 1. f(x)= \sqrt{x- 1} which has domain x\ge 1.

    The domain of fog will be all x, except 1, such that g(x)= \frac{1}{x- 1}\ge 1. If x> 1, x-1> 0 so, multiplying on both sides by the positive number x- 1, 1\ge x- 1 or x\le 2. If x< 1, x- 1< 0 so, multiplying both sides by the negative number x- 1, 1\le x- 1 or x\ge 2. Of course, we can't have "x< 1" and " x\ge 2" so that last case is impossible. We must have x> 1 and x\le 2 as you say.
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