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±3 1.5 Function Notation 47 As long as we substitute numbers other than 3 and −3, the expression r(x) is a real number. Hence,
we write our domain in interval notation as (−∞, −3) ∪ (−3, 3) ∪ (3, ∞). When a formula for a
function is given, we assume the function is valid for all real numbers which make arithmetic sense
when substituted into the formula. This set of numbers is often called the implied domain1 of
the function. At this stage, there are only two mathematical sins we need to avoid: division by 0
and extracting even roots of negative numbers. The following example illustrates these concepts.
Example 1.5.3. Find the domain2 of the following functions.
1. f (x) = 2 4. r(x) = 4
√
6− x+3 5. I (x) = 4x
1−
x−3
√
2. g (x) = 4 − 3x
√
3. h(x) = 5 4 − 3x 3x2
x Solution.
1. In the expression for f , there are two denominators. We need to make sure neither of them is
0. To that end, we set each denominator equal to 0 and solve. For the ‘small’ denominator,
we get x − 3 = 0 or x = 3. For the...

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