if $r(x)=2 - x^{2}$ and $w(x)=x - 2$, what is the range of $(wcirc r)(x)$?\n$(-infty,0$\n$(-infty,2$\n$0,inft…

Explanation:

Step1: Find the composition $(w\circ r)(x)$

$(w\circ r)(x)=w(r(x))$. Since $r(x)=2 - x^{2}$ and $w(x)=x - 2$, then $w(r(x))=(2 - x^{2})-2=-x^{2}$.

Step2: Analyze the function $y = -x^{2}$

The function $y=-x^{2}$ is a parabola. The general form of a parabola is $y = ax^{2}+bx + c$, here $a=-1$, $b = 0$, $c = 0$. The vertex - form of a parabola is $y=a(x - h)^{2}+k$, and for $y=-x^{2}$, the vertex is at $(h,k)=(0,0)$ and $a=-1<0$, so the parabola opens downwards.

Step3: Determine the range

Since the parabola $y=-x^{2}$ opens downwards and its vertex is at $(0,0)$, the maximum value of the function is $y = 0$ and it can take any non - positive value. So the range of $y=-x^{2}$ is $(-\infty,0]$.

Answer:

$(-\infty,0]$