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.

domain and range

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)Screen Shot 2014-03-18 at 8.20.03 PM

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.

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