121616949-math.26 - 12 Chapter 1 Analytic Geometry EXAMPLE...

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12 Chapter 1 Analytic Geometry EXAMPLE 1.5 Find the domain of y = f ( x ) = 1 4 x - x 2 . To answer this question, we must rule out the x -values that make 4 x - x 2 negative (because we cannot take the square root of a negative) and also the x -values that make 4 x - x 2 zero (because if 4 x - x 2 = 0, then when we take the square root we get 0, and we cannot divide by 0). In other words, the domain consists of all x for which 4 x - x 2 is strictly positive. We give two different methods to find out when 4 x - x 2 > 0. First method. Factor 4 x - x 2 as x (4 - x ). The product of two numbers is positive when either both are positive or both are negative, i.e., if either x > 0 and 4 - x > 0, or else x < 0 and 4 - x < 0. The latter alternative is impossible, since if x is negative, then 4 - x is greater than 4, and so cannot be negative. As for the first alternative, the condition 4 - x > 0 can be rewritten (adding x to both sides) as 4 > x , so we need: x > 0 and 4 > x (this is sometimes combined in the form 4 > x > 0, or, equivalently, 0 < x < 4).
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