# Domain and Range on a Graphical Perspective

Two weeks ago, I  discussed the basic concepts of domain and range which I presented in an ‘algebraic way.’ In this post, I would like to discuss these concepts from a graphical perspective.

The domain of a function $x$ is the set of points on the x-axis where if a vertical line is drawn, it will hit a point on the graph. Take for instance, in the linear function $f(x) = 2x$,  we are sure that we can always hit a point wherever we draw a vertical line. In algebraic explanation, we can always find an $f(x)$ for every $x$. Therefore, we can conclude the that domain of $f$ is the set of real numbers. On the other hand, if we draw a horizontal line and it hits the graph, then it is part of the range of the graph. Clearly, the range of the $f$ is also the set of real numbers.

Consider the function in the figure above which we will call $g(x)$. Notice that when we talk about vertical lines, we can only hit the graph if we draw it between $x = 1$ and $x = 3$. Since when we draw a vertical line passing through any of these points we hit the graph, we can say that the domain of $g(x)$ is the set of real numbers between $1$ and $3$. The range of $g(x)$ is represented by the green line segment on the y-axis. Again, if we draw a vertical line passing any of those points, it will pass the graph. So, the range of $g(x)$ is between 3.1 and 5.8 (approximately).

The next graph below which we will call $h(x)$ is just the same as the first figure only that it has a “hole.” The hole there means that the point on the graph does not exist. So, if the x-coordinate of that whole is, for example, 2.5, then we can say that $x = 2.5$ is not part of the domain of $h(x)$. Therefore the domain of $h(x)$ is the set of all points between 1 and 3 except 2.5. As you would surely guess $h(2.5)$ is not part of the range of $h(x)$

In  this post and in the previous, we have learned that the domain of a function is the set for which makes the range defined. In graphical representation, the domain of the function f is the set of x’s where $f(x)'s$ exist. It means that with in that interval, the graph “above” exist.