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Math Help - Some clarification on ranges when combining functions

  1. #1
    Member alexgeek's Avatar
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    Some clarification on ranges when combining functions

    A question I just tried has me thinking about the range of a composite function:

    <br />
\begin{array}{lll}<br />
f: x\to x-\frac{1}{x} & [1, \infty) & [0, \infty) \\<br />
g: x\to 3x^2 + 2 & [0, \infty) & [2, \infty) \\<br />
\end{array}<br />

    Find <br />
gf(x) = \frac{3}{x^2}+7 <br />

    <br />
\begin{array}{l}<br />
3(x-\frac{1}{x})^2 + 2 = \frac{3}{x^2} + 8 \\<br />
3x^2 = 12 \\<br />
x = \pm 2<br />
\end{array}<br />

    The range of g is [2,\infty) though, so is the only answer x = 2 or is x=-2 also an answer?

    Thanks
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  2. #2
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    Since you restricted the domain of f, it is not permitted to substitute -2 in for x in the function f. So x=-2 cannot be a solution to the given equation.
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  3. #3
    Member alexgeek's Avatar
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    Quote Originally Posted by DrSteve View Post
    Since you restricted the domain of f, it is not permitted to substitute -2 in for x in the function f. So x=-2 cannot be a solution to the given equation.
    Ah thank you, I was thinking only in terms of the range and not the domain.
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