SECTION A.1 Algebra Essentials A7 5 Determine the Domain of a Variable In working with expressions or formulas involving variables, the variables may be allowed to take on values from only a certain set of numbers. For example, in the formula for the area A of a circle of radius r , A r ,2 π = the variable r is necessarily restricted to the positive real numbers. In the expression x 1 , the variable x cannot take on the value 0, since division by 0 is not defined. Finding the Domain of a Variable The domain of the variable x in the expression x 5 2 − is x x 2 { } ≠ since, if x 2, = the denominator becomes 0, which is not defined. EXAMPLE 8 Circumference of a Circle In the formula for the circumference C of a circle of radius r , C r 2π = the domain of the variable r , representing the radius of the circle, is the set of positive real numbers, r r 0 . { } > The domain of the variable C , representing the circumference of the circle, is also the set of positive real numbers, C C 0 . { } > EXAMPLE 9 (b) If x 3 = and y 1, = − then xy 5 5 3 1 15 ( ) = ⋅ ⋅ − = − (c) If x 3 = and y 1, = − then y x 3 2 2 3 1 2 2 3 3 2 6 3 4 3 4 ( ) − = − − ⋅ = − − = − − = (d) If x 3 = and y 1, = − then x y 4 4 3 1 12 1 13 13 ( ) ( ) − + =−⋅+− =− +− =− = Now Work problems 51 and 59 DEFINITION Domain of a Variable The set of values that a variable may assume is called the domain of the variable . In describing the domain of a variable, we may use either set notation or words, whichever is more convenient. Now Work problem 69 6 Use the Laws of Exponents Integer exponents provide a shorthand notation for representing repeated multiplications of a real number. For example, 3 3 3 3 3 81 4 = ⋅ ⋅ ⋅ =
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