Finding the range of a function without graph? 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 : https://www.khanacademy.org/math/algebra/algebra-functions/domain-and-range/v/domain-and-range-from-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 ?
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 $(-\infty, 1]$.
You see that if we put x as infinity we get f(x)=-∞.
Now to find the maximum value of functions just differentiate it and put the derivative equal to zero to find the point where the function stops increasing and starts decreasing.
$\frac{d}{dx}(-x^2 + 1) = -2x$
$-2x=0 =&gt;$ f(x) is maximum when x=0
Therefore maximum value of f(x) is +1
$f(x)\epsilon (-∞,+1) $ here is the max value on graph