Chapter 1 | Functions and Graphs
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1.11 Find the domain and range for the function f ( x ) = (5 x +2)/(2 x −1).
The root functions f ( x ) = x 1/ n have defining characteristics depending on whether n is odd or even. For all even integers n ≥2, the domain of f ( x ) = x 1/ n is the interval [0, ∞). For all odd integers n ≥1, the domain of f ( x ) = x 1/ n is the set of all real numbers. Since x 1/ n = (− x ) 1/ n for odd integers n , f ( x ) = x 1/ n is an odd function if n is odd. See the graphs of root functions for different values of n in Figure 1.21 .
Figure 1.21 (a) If n is even, the domain of f ( x ) = x n is [0, ∞). (b) If n is odd, the domain of f ( x ) = x n is (−∞, ∞) and the function f ( x ) = x n is an odd function.
Example 1.17 Finding Domains for Algebraic Functions
For each of the following functions, determine the domain of the function. a. f ( x ) = 3 x 2 −1 b. f ( x ) = 2 x +5 3 x 2 +4
c. f ( x ) = 4−3 x d. f ( x ) = 2 x −1 3
Solution a. You cannot divide by zero, so the domain is the set of values x such that x 2 −1≠0. Therefore, the domain is { x | x ≠±1}. b. You need to determine the values of x for which the denominator is zero. Since 3 x 2 +4≥4 for all real numbers x , the denominator is never zero. Therefore, the domain is (−∞, ∞). c. Since the square root of a negative number is not a real number, the domain is the set of values x for
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