I've found the domain but how do we find the range and sketch graph. is there a way to find the range. Also the graph of square root is like $e^x$ but facing downwards towards $x$ axis while $x^2$ is a parabola and $9$ units upwards. I just don't get the graph.
[Math] find the domain and range and sketch the graph of $g(x) = \sqrt{9 – x^2}$.
algebra-precalculus
Best Answer
For a function $g(x) = \sqrt{9 - x^2},$ we can square it to find $g^2(x) = 9 - x^2.$
Adding $x^2$ to both sides, we now have $g^2(x) + x^2 = 9.$ Replace $g(x) = y$ to get $x^2 + y^2 = 9$.
We know that $x^2 + y^2 = r^2$ is the equation for a circle of radius $r,$ so we'll sketch a circle of radius $3$ (because $3^2 = 9$) centered around the origin.
However, since in our original problem it was a square root, we'll assume the positive root and sketch only the top half of a circle.
The domain and range can be found, respectively, as $\left[-3, 3\right]$ and $\left[0, 3\right],$ which can be determined from the sketch.
Ask as many questions as you can, as my problem solving isn't necessarily spot-on. ;-)