use the graph shown to find the following. (a) the domain and range of the function (b) the intercepts, if…

Explanation:

Step1: Determine domain

The function is defined for all real - x values except where the denominator is zero. From the graph, there are no breaks in the x - direction that would limit the domain. So the domain is all real numbers, $(-\infty,\infty)$.

Step2: Determine range

The graph has no upper or lower bounds. The function can take on any real - y value. So the range is all real numbers, $(-\infty,\infty)$.

Step3: Find x - intercepts

The x - intercepts are the points where the graph crosses the x - axis. From the graph, the x - intercepts are $x=-1$ and $x = 1$.

Step4: Find y - intercepts

The y - intercept is the point where the graph crosses the y - axis. Since the graph is symmetric about the origin and passes through $(-1,0)$ and $(1,0)$, when $x = 0$, the function value is $y = 0$. So the y - intercept is $y = 0$.

Step5: Check horizontal asymptotes

As $x\to\pm\infty$, the function does not approach a constant value. So there are no horizontal asymptotes.

Step6: Check vertical asymptotes

The graph does not have any vertical lines where the function approaches $\pm\infty$. So there are no vertical asymptotes.

Step7: Check oblique asymptotes

As $x\to\pm\infty$, the function does not approach a non - horizontal, non - vertical line. So there are no oblique asymptotes.

Answer:

(a) Domain: $(-\infty,\infty)$; Range: $(-\infty,\infty)$ (b) x - intercepts: $x=-1,x = 1$; y - intercept: $y = 0$ (c) None (d) None (e) None