SECTION 4.5 Properties of Rational Functions 237 If ( ) ( ) ( ) = R x p x q x is a rational function, and if p and q have no common factors, then the rational function R is said to be in lowest terms. For a rational function ( ) ( ) ( ) = R x p x q x in lowest terms, the real zeros, if any, of the numerator, which are also in the domain of R, are the x-intercepts of the graph of R. The real zeros of the denominator of R [that is, the numbers x, if any, for which ( ) = q x 0], although not in the domain of R, also play a major role in the graph of R. We have already discussed the properties of the rational function = y x 1 . (Refer to Example 9, page 25). The next rational function that we analyze is ( ) = H x x 1 . 2 Graphing ( ) = H x x 1 2 Analyze the graph of ( ) = H x x 1 . 2 EXAMPLE 2 Figure 35 ( ) = H x x 1 2 Figure 36 ( ) = H x x 1 2 x x 5 0 y 5 0 y 5 0 3 23 y 5 2, (1, 1) 1 – 4 , 4 1 – 2 (22, (21, 1) 1 – 4 (2 , 4 1 – 2 ) ( ) ( ) ) Solution The domain of ( ) = H x x 1 2 consists of all real numbers x except 0. The graph has no y-intercept, since x can never equal 0. The graph has no x-intercept because the equation ( ) = H x 0 has no solution.Therefore, the graph of H does not cross either coordinate axis. Because ( ) ( ) ( ) − = − = = H x x x H x 1 1 2 2 H is an even function, so its graph is symmetric with respect to the y-axis. Figure 35 shows the graph of H using Desmos. Notice that the graph confirms the conclusions just reached. But what happens to the graph as the values of x get closer and closer to 0? We use a table to answer the question; see Table 11 obtained from Desmos. The first four rows show that as the values of x approach (get closer to) 0, the values of ( ) H x become unbounded in the positive direction. That is, as ( ) → →∞ x H x 0, . In calculus, we use limit notation, ( ) =∞ → H x lim , x 0 to convey this. Look at the last four rows of Table 11. As →∞ x , then ( ) → H x 0 (the end behavior of the graph). This is expressed in calculus by writing ( ) = →∞ H x lim 0. x Remember, on the calculator − 1E 4 means × − 1 10 4 or 0.0001. Figure 36 shows the graph of ( ) = H x x 1 2 drawn by hand. Notice the use of red dashed lines to convey these ideas. Table 11 Table 10 CAUTION The domain of a rational function must be found before writing the function in lowest terms. j
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