Answer:
The first choice, [tex](-\infty, 7) \cup (7, \infty)[/tex].
Step-by-step explanation:
The dashed vertical line is a vertical asymptote. The x-coordinate of all points on this vertical asymptote are equal to 7. In other words, when the x-value of a point on the graph approaches [tex]7[/tex], its y-value approaches infinity (or negative infinity.)
Either way, the graph is not defined for [tex]x = 7[/tex]. The point [tex]7[/tex] should thus be excluded from the domain of the graph.
The graph is apparently defined for all other x-values. The domain of the function should thus be all real numbers with the exception of [tex]x = 7[/tex]. Here's how to write that in interval notation:
- The set of all real numbers can be expressed as the interval [tex](-\infty, \infty)[/tex]. Note that infinity [tex]\infty[/tex] (or [tex](- \infty)[/tex] itself isn't a real number. The round brackets indicate that both endpoints are excluded from the from the interval.
- The set of all real numbers less than (not equal to) [tex]7[/tex] is [tex](-\infty,7 )[/tex] (both ends are excluded.) The set of all real numbers greater than (not equal to) [tex]7[/tex] is [tex](7, \infty)[/tex].
- The set of all real numbers that is not equal to [tex]7[/tex] is the union of all real numbers less than [tex]7[/tex] and all real numbers greater than [tex]7[/tex]. The union operator [tex]\cup[/tex] connects two intervals.
In other words, the domain is [tex](-\infty, 7) \cup (7, \infty)[/tex].
