Question

a graph of the function f is given below: study the graph of f at the value given below. select each of the following that applies for the value a = -6. f is defined at x = a. f is not defined at x = a. f is continuous at x = a. f is discontinuous at x = a. f is smooth at x = a. f is not smooth at x = a. f has a horizontal tangent line at x = a. f has a vertical tangent line at x = a. f has a oblique/slanted tangent line at x = a. f has no tangent line at x = a.

Explanation:

Step1: Check if function is defined

Locate $x = - 6$ on the x - axis. The graph has a point at $x=-6$, so $f$ is defined at $x = a=-6$.

Step2: Check continuity

The graph has a break at $x = - 6$, so $f$ is discontinuous at $x = a=-6$.

Step3: Check smoothness

Since the function is discontinuous at $x=-6$, it is not smooth at $x = - 6$.

Step4: Check tangent line

Because of the discontinuity at $x=-6$, there is no well - defined tangent line at $x=-6$.

Answer:

$f$ is defined at $x = a$.
$f$ is discontinuous at $x = a$.
$f$ is not smooth at $x = a$.
$f$ has no tangent line at $x = a$.