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Domain and Range

We usually identify a function with its correspondence, as in the example above. However, a function consists of three parts: a domain, a range, and correspondence between them.


The domain of a function is the set of x values for which the function is defined.
For example, the function  is defined for all values of x ≠ 1, which causes division by zero. There is an infinite variety of functions with restricted domains, but only two types of restricted domains appear on the SAT: division by zero and even roots of negative numbers.
For example, the function  is defined only if x – 2 ≥ 0, or x ≥ 2. The two types of restrictions can be combined.
For example, . Here, x – 2 ≥ 0 since it’s under the square root symbol. Further x – 2 ≠ 0, or x ≠ 2, because that would cause division by zero. Hence, the domain is all x > 2.
The range of a function is the set of y values that are assigned to the x values in the domain.

For example, the range of the function y = f(x) = x2 is y ≥ 0 since a square is never negative. The range of the function y = f(x) = x2 + 1 is y ≥ 1 since x2 + 1 ≥ 1. You can always calculate the range of a function algebraically, but it is usually better to graph the function and read off its range from the y values of the graph.

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