I have a quarter 1 test tomorrow and I have looked on many websites containing set builder notation to find domain and range but I still don't understand. Is there anyway to remember it or even how to do it??
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$\begingroup$If you're given a graph the range is all of the $y$ values and the domain is all of the $x$ values where the graph exists.
For example, consider this graph
What are the $x$ values at which the function is defined? Well we can see it starts at $-3$ on the left and keeps going until $4$. Notice, that even though this is a piecewise function, every single $x$ between $-3$ and $4$ corresponds to a point on the graph. Then we just need to take into a account whether the endpoints are included or not. In this case $-3$ is but $4$ is not. So the domain, in set builder notation, is $\{x\mid -3\le x\lt 4\}$.
As for the range, we look at the $y$ values. The lowest $y$ value at which the function is defined is $-3$. Then continuing up we see a break from $0$ to $1$. There is no point on the graph that corresponds to $y$ values between those two numbers. But then it continues at $1$ and goes up to $2$. In this case $-3$, $0$, and $1$ are definitely included. It might be slightly harder to tell that $2$is included, but it is. So the range is $\{y \mid -3\le y\le 0 \text{ or } 1\le y \le 2\}$.
$\endgroup$ 2 $\begingroup$I'm going to make some assumptions, let me know if this is what you have in mind.
Suppose we are given this graph:
and we are asked to give the domain and range of the function in set builder notation.
We can see that the function has a value at all points except where $x = 0$. So the domain of the function is $\{ x \in \mathbb{R} : x \neq 0 \} $
Now we will consider the range. Note that the function goes off to $\pm\infty$ near the origin, so most values are in the range of the function. However, this particular function never crosses the x-axis, so 0 is not in the range. Therefore, the range is $\{ y \in \mathbb{R} : y \neq 0 \}$.
In all questions of this form, you have to first: identify the domain and range, and second: write it in set-builder notation. You can think about finding the range by imagining horizontal lines and seeing at what y-values they do (and do not) intersect the graph. Likewise with horizontal lines for the domain.
$\endgroup$ $\begingroup$Going off the comment, I assume you want to know how to give the domain and range of a function using set builder notation. For a given function $f(x)$, you can think of the domain as everything that you can put into the function, and the range as everything that you can get out of the function. For example, if we have a line of the form $y = 5x + 2$, then the domain would be all the possible values you can put in for $x$, and the range would be all output $y$ values. Now, in this case, the domain is all real numbers, as you can put in any $x$, and the output is also all reals, as $y$ can take on any value on the real line. In this case, the domain and range would just be $\mathbb{R}$, the set of all real numbers.
Consider a different function like $y = \sqrt{x}$. The square root is not defined for negative numbers, so we have to restrict the domain. So the domain would be $\{x : x \geq 0\}$. Similarly, the range is only non-negative numbers, so the range would be the same set. The idea is that you can use this notation to describe precisely the set of possible "inputs" and "outputs" for your given function.
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