I'm asked to find the range of the function :
$ (-x^2 , +1) $
Taking R as a real number which represents a quantity along a line and graphing :
then the range for this function from viewing the graph appears to be :
$ R = (-\infty , +\infty) $
Is there an alternative method of finding the range of this function without using a graph ?
Watching the khan academy tutorial suggests using graphs : but is there a pure algebraic method instead of using graphs ?
It is not clear what the range is when the range appears infinite as how do we know that at some point on the axis the range functions stops tending towards infinity ?
$\endgroup$1 Answer
$\begingroup$Notice that $x^2 \ge 0$ for all $x \in \Bbb R$, which means that $1 - x^2 \le 1$ for all $x \in \Bbb R$, with equality for $x=0$. Notice, too, that $x^2$ increases towards $\infty$, therefore $1-x^2$ will decrease towards $-\infty$. Finally, notice that $x^2$ is surjective (for every $y \ge 0$, there exist $\sqrt y$). Therefore, the values of $f$ lay between $- \infty$ and $1$, meaning that the range is $(-\infty, 1]$.
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