identifying the graph of an inverse trig function\nwhich is the graph of arctan(x)?

Answer:

The graph of $y = \arctan(x)$ is a smooth - increasing curve that has horizontal asymptotes at $y=-\frac{\pi}{2}$ and $y = \frac{\pi}{2}$. It passes through the origin $(0,0)$. Among the given graphs, the one that has these characteristics is the second graph (assuming the graphs are numbered from left - to - right).

Explanation:

Step1: Recall domain and range

The domain of $y=\arctan(x)$ is $(-\infty,\infty)$ and the range is $(-\frac{\pi}{2},\frac{\pi}{2})$.

Step2: Analyze end - behavior

As $x\to-\infty$, $y=\arctan(x)\to-\frac{\pi}{2}$ and as $x\to\infty$, $y=\arctan(x)\to\frac{\pi}{2}$.

Step3: Check for passing through origin

When $x = 0$, $\arctan(0)=0$, so the graph passes through $(0,0)$.

Step4: Identify correct graph

Based on domain, range, end - behavior and passing through origin, we can identify the correct graph.