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Thread: Range of Composite Functions

  1. #1
    Senior Member Stroodle's Avatar
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    Range of Composite Functions

    Hi.

    If:
    $\displaystyle f:R\rightarrow R,\;f(x)=x^2-4$
    and
    $\displaystyle g:R+\cup\left\{0\right\}\rightarrow R,\;g(x)=\sqrt x$

    Then what is the range of $\displaystyle f\circ g$ and why?

    And what's the latex command for the little + symbol that normally goes next to R in the domain?

    Thanks for your help.
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  2. #2
    Senior Member pankaj's Avatar
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    $\displaystyle
    fog=(g(x))^2-4=(\sqrt x)^2-4\geq 0-4\geq -4
    $

    Thus range$\displaystyle =(-4,\infty]$
    Last edited by pankaj; Jun 17th 2009 at 08:01 AM.
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  3. #3
    Senior Member Stroodle's Avatar
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    Thanks heaps for your reply, but I don't get that, and my text says the answer is $\displaystyle [-4,\infty)$
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  4. #4
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    Quote Originally Posted by Stroodle View Post
    Hi.

    If:
    $\displaystyle f:R\rightarrow R,\;f(x)=x^2-4$
    and
    $\displaystyle g:R+\cup\left\{0\right\}\rightarrow R,\;g(x)=\sqrt x$

    Then what is the range of $\displaystyle f\circ g$ and why?
    g(x) maps the non-negative real numbers to the set of non-negative real numbers: given any $\displaystyle y\ge 0$, take $\displaystyle x= y^2$ so y= g(x).
    Now apply f(x) to such a number. If $\displaystyle x\ge 0$, then $\displaystyle x^2\ge 0$ so $\displaystyle x^2- 4\ge -4$. The range of $\displaystyle f\circ g(x)$ is all $\displaystyle x\ge -4$.

    And what's the latex command for the little + symbol that normally goes next to R in the domain?
    $\displaystyle R^+$ is the set of positive real numbers. Adding 0 to that set gives the set of "non-negative" real numbers. You need that for the domain of g because you cannot take the square root of a negative number.

    Thanks for your help.
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  5. #5
    Senior Member Stroodle's Avatar
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    Awesome. Thankyou
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  6. #6
    Senior Member pankaj's Avatar
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    Quote Originally Posted by Stroodle View Post
    Thanks heaps for your reply, but I don't get that, and my text says the answer is $\displaystyle [-4,\infty)$
    Yor text is right.I have edited my post.
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