How do I identify the domain and range by looking at a graph?

Expert Answers
mvcdc eNotes educator| Certified Educator

Domain is the set of inputs for which the function is defined - that is, these are all the 'valid' values that we can plug in to our function. Range, on the other hand, would be all sets of values that we could possibly get by plugging in all values in the domain. 

For instance, if you have a function `f(x) = 1/x` , then 5 would be part of its domain since `1/5` is defined, but 0 won't be since `1/0` is undefined. Similarly, `1/5` is part of the range, but 0 is not since there is no way we can possibly get 0 from the given function. ``

If you noticed, we are actually getting 'points' on a cartesian plane when plugging in values and solving for f(x) -- the value we plugged in is the x-coordinate, while the value we eventually get is the y-coordinate. Hence, if we want to identify the domain and range from a graph (which is a lot easier), we simply have to examine the graph, and pick out the areas in which the curve (of the function) actually passes. 

For example, say you have a function f(x) = x. This is obviously just a straight line -- in fact one that passes through the origin, containing all points for which x = y. Now, we know that a line extends to infinity -- it never ends. Hence, at all possible x values we can think of, there will always be a point in the line corresponding to that value, and same is true for the case of y.

I have attached two other graphs for examples.

The first example is the graph of `y=x^2` . Here, it is obvious that the graph doesn't have any part of it that goes to the negative y-axis (no part of the graph in the 3rd and 4th quadrants). Hence, negative numbers aren't part of its range. However, the curve extends to infinity (in the positive y-axis). Hence, its range is all values of y greater than or equal to zero: `R: y >= 0` . For any x value that we pick, on the other hand, there's a part of the graph corresponding to that point. If you notice, the branches of the graph extend both to positive and negative infinity. Hence, the domain is all real numbers.

The second graph is the graph of `y=1/x` . An obvious observation here would be that it never touches the y-axis and the x-axis. This means that x=0 cannot be plugged in, and y=0 cannot ever be computed from the function. However, the rest of the graph extends to positive infinity and negative infinity, both along the x- and y-axes. Hence, the domain and range would be all real numbers, except for 0: `D: x ne 0`and `R:yne0`  .

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