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

Explanation:

Step1: Identify domain from graph

The function exists for all real - x values except where there is a vertical asymptote. The vertical asymptote is at x = - 6. So the domain is all real numbers except x=-6. $(-\infty,-6)\cup(-6,\infty)$

Step2: Identify range from graph

The function exists for all real - y values except where there is a horizontal asymptote. The horizontal asymptote is at y = 2. So the range is all real numbers except y = 2. $(-\infty,2)\cup(2,\infty)$

Step3: Find x - intercept

The graph crosses the x - axis at x = 0. So the x - intercept is 0.

Step4: Find y - intercept

The graph crosses the y - axis at y = 0. So the y - intercept is 0.

Step5: Identify horizontal asymptote

As x approaches positive or negative infinity, the function approaches y = 2. So the horizontal asymptote is y = 2.

Step6: Identify vertical asymptote

The function has a break at x=-6, and the graph approaches infinity or negative infinity as x approaches - 6. So the vertical asymptote is x=-6.

Step7: Identify oblique asymptote

Since the degree of the numerator is not one more than the degree of the denominator (from the behavior of the graph indicating a rational - like function), there is no oblique asymptote.

Answer:

(a) Domain: $(-\infty,-6)\cup(-6,\infty)$ (b) Intercepts: x - intercept: 0, y - intercept: 0 (c) Horizontal asymptote: y = 2 (d) Vertical asymptote: x=-6 (e) Oblique asymptote: None