381 3.5 Rational Functions: Graphs, Applications, and Models As x increases without bound (written x S∞), the values of ƒ1x2 = 1 x get closer and closer to 0, as shown in the tables in Figure 42. Letting x increase without bound causes the graph of ƒ1x2 = 1 x to move closer and closer to the horizontal line y = 0. This line is a horizontal asymptote. ƒ1x2 = 1 x , ƒ1x2 = x + 1 2x2 + 5x - 3 , ƒ1x2 = 3x2 - 3x - 6 x2 + 8x + 16 Rational functions Any values of x such that q1x2 = 0 are excluded from the domain of a rational function, so this type of function often has a discontinuous graph—that is, a graph that has one or more breaks in it. As x approaches 0 from the left, y1 = 1 x approaches -∞. (–1E–6 means -1 * 10-6.) As x approaches 0 from the right, y1 = 1 x approaches ∞. Figure 41 As x approaches -∞, y1 = 1 x approaches 0 through negative values. As x approaches ∞, y1 = 1 x approaches 0 through positive values. Figure 42 The Reciprocal Function f 1x2 =1 x The simplest rational function with a variable denominator is the reciprocal function. ƒ1x2 = 1 x Reciprocal function The domain of this function is the set of all nonzero real numbers. The number 0 cannot be used as a value of x, but it is helpful to find values of ƒ1x2 for some values of x very close to 0. We use the table feature of a graphing calculator to do this. The tables in Figure 41 suggest that ƒ1x2 increases without bound as x gets closer and closer to 0, which is written in symbols as ƒ1x2 S∞ as x S0. (The symbol x S0 means that x approaches 0, without necessarily ever being equal to 0.) Because x cannot equal 0, the graph of ƒ1x2 = 1 x will never intersect the vertical line x = 0. This line is a vertical asymptote.
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