I have been given the following relations to find the domain and range of using builder notation. (The blue writing is what I have so far)

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I am just beginning to learn the whole concept of set builder notation, and I am running into a little confusion. I understand the x and y axis, as well as the form it is written in. I'm confused with question b and c, because of the arrow end points that 'keep going'.


1 Answer 1


You're on the right track, but you're having trouble with the endpoints.

The points $(-3, -2)$ and $(4, -2)$ are on the graph in a), which means that your domain should actually be $x \in [-3, 4]$ in interval notation to indicate that $x = -3$ and $x = 4$ are included. This would be reflected in the set-builder notation by using inequalities with "or equal to", so that you'd have $\{x \in \Bbb R \mid -3 \le x \le 4\}$.

For part c), you're again almost right, except for the endpoints. Their $x$- and $y$-coordinates need to be included, so you'd need brackets for interval notation, and all $\le$'s for the set-builder notation. So, for the range, I would write $\{y \in \Bbb R \mid -3 \le y \le 3\}$, using $y$ rather than $x$.

Now, in addition to filled-in circles, some graphs have arrows. These indicate that the graph "keeps going" in (roughly) whatever direction the arrows point. In b), this would be reflected as an interval $x \in (-\infty, 3]$ for the domain.

Notice for b) that $x$ needs to only be "at most $3$" (not "at least" anything), and thus you'll only need a single inequality, rather than the compound ones you would use on graphs that have a definite starting and ending point.

I'll let you give the rest a shot; most of what you had was spot-on.

  • $\begingroup$ I think I'm starting to understand how to use builder set notation but I think I'm still struggling with the arrows in questions b and d. This is what I have so far. A) D: { x | -3 ≤ x ≤ 4, x ∈ ℝ } R: { y | -2 ≤ y ≤ 4, y ∈ ℝ } B) D: { x | x < 3, x ∈ ℝ } R: { y | y < 3, y ∈ ℝ } C) D:{x | -3 ≤ x ≤ 4, x ∈ ℝ } R:{y | -3 ≤ y ≤ 3, y ∈ ℝ } D) D: {x | x < 4, x ∈ ℝ } R: { y | y < 3, x ∈ ℝ } $\endgroup$
    – Madison
    Aug 3, 2015 at 3:19

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