Question

Which is the interval notation represent the domain?

Answer 1

The first choice,
`(-\infty, 7) \cup (7, \infty)`.

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
`7`, its y-value approaches infinity (or negative infinity.)

Either way, the graph is not defined for
`x = 7`. The point
`7` 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
`x = 7`. Here's how to write that in interval notation:

• The set of all real numbers can be expressed as the interval
  `(-\infty, \infty)`. Note that infinity
  `\infty` (or
  `(- \infty)` 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)
  `7` is
  `(-\infty,7 )` (both ends are excluded.) The set of all real numbers greater than (not equal to)
  `7` is
  `(7, \infty)`.
• The set of all real numbers that is not equal to
  `7` is the union of all real numbers less than
  `7` and all real numbers greater than
  `7`. The union operator
  `\cup` connects two intervals.

In other words, the domain is
`(-\infty, 7) \cup (7, \infty)`.