70 CHAPTER 2 Functions and Their Graphs The following may prove helpful for finding the domain of a function that is defined by an equation and whose domain is a subset of the real numbers. Finding the Domain of a Function Find the domain of each function. (a) ( ) = + f x x x5 2 (b) ( ) = − g x x x 3 4 2 (c) ( ) = − h t t 4 3 (d) ( ) = + − F x x x 3 12 5 Solution EXAMPLE 9 (a) The function ( ) = + f x x x5 2 says to sum the square of a number and five times the number. Since these operations can be performed on any real number, the domain of f is the set of all real numbers. (b) The function ( ) = − g x x x 3 4 2 says to divide x3 by − x 4. 2 Since division by 0 is not defined, the denominator − x 4 2 cannot be 0, so x cannot equal −2 or 2. The domain of the function g is { } ≠ − ≠ x x x 2, 2 . (c) The function ( ) = − h t t 4 3 says to take the square root of − t 4 3 . Since only nonnegative numbers have real square roots, the expression under the square root (the radicand) must be nonnegative (greater than or equal to zero). That is, − ≥ − ≥− ≤ t t t 4 3 0 3 4 4 3 The domain of h is { } ≤ t t 4 3 , or the interval (−∞ ⎤ ⎦ ⎥ , 4 3 . (d) The function ( ) = + − F x x x 3 12 5 says to take the square root of +x3 12 and divide the result by −x 5. This requires that + ≥ x3 12 0, so ≥− x 4, and also that − ≠ x 5 0, so ≠ x 5. Combining these two restrictions, the domain of F is { } ≥− ≠ x x x 4, 5 Finding the Domain of a Function Defined by an Equation • Start with the domain as the set of all real numbers. • If the equation has a denominator, exclude any numbers for which the denominator is zero. • If the equation has a radical with an even index, exclude any numbers for which the expression inside the radical (the radicand) is negative. To find the domain, solve ≥ radicand 0. Now Work PROBLEM 55 We express the domain of a function using interval notation, set notation, or words, whichever is most convenient. If x is in the domain of a function f, we say that f is defined at x , or ( ) f x exists . If x is not in the domain of f, we say that f is not defined at x , or ( ) f x does not exist . For example, if ( ) = − f x x x 1 , 2 then ( ) f 0 exists, but ( ) f 1 and ( ) − f 1 do not exist. (Do you see why?) When a function is defined by an equation, it can be difficult to find its range unless we also have a graph of the function. So we are usually content to find only the domain. In Words The domain of g found in Example 9(b) is { } ≠− ≠ x x x 2, 2 . This notation is read, “The domain of the function g is the set of all real numbers x such that x does not equal −2 and x does not equal 2.”
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