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In the equation $f(x)=-x^2 +2x -7$, I am told the range is $(-\infty,-6]$. I feel the domain should be $(-\infty, \infty)$ because we can put any value in for $x$ and still get a real number. If the domain is $(-\infty,\infty)$, what makes the range $(-\infty, -6]$? I don't get it. Is there a more intuitive way for me to understand this? Thanks.

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Have you tried drawing a graph of $f$? – Trevor Wilson Sep 6 '12 at 2:32
Can you recite your definition of range? – Alex Becker Sep 6 '12 at 2:34
up vote 3 down vote accepted

The range is the set of values the function can take. You are correct the domain is all of $\mathbb R$. If you plot this, it is a parabola opening downward and $-6$ is the vertex. You can see this by writing $f(x)=-(x-1)^2-6.$ The squared term has a maximum value of $0$, so the the maximum of $f(x)$ is $-6$.

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That helped tremendously. – Kyle H Sep 6 '12 at 2:49

You are correct about the domain. The range is simply all the possible values that $f(x)$ can take. If you graph $y=f(x)$ you will see that it is an "upside-down" parabola. Therefore it has a maximum $y$ value which happens to be -6.

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Thanks for the help. – Kyle H Sep 6 '12 at 2:53

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