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# Sets and Functions Help (page 2)

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By — McGraw-Hill Professional
Updated on Sep 1, 2011

#### Example 2

Let . Determine a domain and range for f which make f a function.

#### Solution 2

Notice that f makes sense for x [−1, 1] (we may not take the square root of a negative number, so we cannot allow x > 1 or x < −1). If we understand f to have domain [−1, 1] and range , then f : [−1, 1] → is a function.

Math Note: When a function is given by a formula, as in Example 1.25, with no statement about the domain, then the domain is understood to be the set of all x for which the formula makes sense.

You Try It: Let

What are the domain and range of this function?

#### Example 3

Let

Determine whether f is a function.

#### Solution 3

Notice that f unambiguously assigns to each real number another real number. The rule is given in two pieces, but it is still a valid rule. Therefore it is a function with domain equal to and range equal to . It is also perfectly correct to take the range to be (−4, ∞), for example, since f only takes values in this set.

Math Note: One point that you should learn from this example is that a function may be specified by different formulas on different parts of the domain .

You Try It: Does the expression

define a function? Why or why not?

#### Example 4

Let . Discuss whether f is a function.

#### Solution 4

This f can only make sense for x ≥ 0. But even then f is not a function since it is ambiguous. For instance, it assigns to x = 1 both the numbers 1 and −1.

Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.

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