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Math Help - Range of composite functions

  1. #1
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    Range of composite functions

    Given two functions:

    f(x)=sqrt(4-x)
    g(x)=x^2

    Find the range of gof:
    gof(x)=4-x

    Shouldn't the range be all real numbers? Because it is a linear function. But the answer key says [0,infinity[ ??

    Next question:

    Given two functions:
    f(x)=1/x (x cannot be 0)
    g(x)1/x^2 (x cannot be 0)

    Find the range of fog

    fog(x)=x^2

    Shouldn't the range by [0,infinity[ ?? The answer key says ]0,infinity[

    Can someone please help me out -3-, I am so frustrated...
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  2. #2
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    Re: Range of composite functions

    f(x)=sqrt(4-x)
    g(x)=x^2

    Find the range of gof:
    gof(x)=4-x

    Shouldn't the range be all real numbers? Because it is a linear function. But the answer key says [0,infinity[ ??
    If the range is all real numbers, for which x do we have g(f(x)) = -1? Be sure to calculate g(f(x)) by applying f to x and then g to the result, not 4 - x to x. Concerning "gof(x)=4-x", recall that a function includes not just a way to convert an input into an output, but also a domain. The fact that two functions are equal implies that their domains are the same.

    Given two functions:
    f(x)=1/x (x cannot be 0)
    g(x)1/x^2 (x cannot be 0)

    Find the range of fog

    fog(x)=x^2

    Shouldn't the range by [0,infinity[ ?? The answer key says ]0,infinity[
    If the range is [0,infinity[, for which x do we have f(g(x)) = 0? Again, forget fog(x)=x^2 and calculate f(g(x)) by definition for the specific x that you think should work.
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  3. #3
    MHF Contributor

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    Re: Range of composite functions

    Quote Originally Posted by kevinlam2490 View Post
    Given two functions:

    f(x)=sqrt(4-x)
    g(x)=x^2

    Find the range of gof:
    gof(x)=4-x
    This is not exactly correct. f(x) exists only if x\le 4. Therefore, gof(x)= 4- x with domain x\le 4, not the same function as "4- x" (for all real x). Given that x cannot be larger than 4, 4- x cannot be less than 0.

    Shouldn't the range be all real numbers? Because it is a linear function. But the answer key says [0,infinity[ ??

    Next question:

    Given two functions:
    f(x)=1/x (x cannot be 0)
    g(x)1/x^2 (x cannot be 0)

    Find the range of fog

    fog(x)=x^2

    Shouldn't the range by [0,infinity[ ?? The answer key says ]0,infinity[

    Can someone please help me out -3-, I am so frustrated...
    But you said "x cannot be 0" so that [itex]x^2[/itex] is never 0 because x cannot be 0. You are not paying enough attention to the domain of the composite function. A "function", even when a formula is given, is not just the formula- it is a specific domain and a formula.

    That is, " f(x)= x^2" and " g(x)= x^2 for x not equal to 0" are different functions!
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