Chapter 1 | Functions and Graphs
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Figure 1.18 (a) For any even integer n , f ( x ) = ax n is an even function. (b) For any odd integer n , f ( x ) = ax n is an odd function.
Behavior at Infinity To determine the behavior of a function f as the inputs approach infinity, we look at the values f ( x ) as the inputs, x , become larger. For some functions, the values of f ( x ) approach a finite number. For example, for the function f ( x ) =2+1/ x , the values 1/ x become closer and closer to zero for all values of x as they get larger and larger. For this function, we say “ f ( x ) approaches two as x goes to infinity,” and we write f ( x )→2 as x →∞. The line y =2 is a horizontal asymptote for the function f ( x ) =2+1/ x because the graph of the function gets closer to the line as x gets larger. For other functions, the values f ( x ) may not approach a finite number but instead may become larger for all values of x as they get larger. In that case, we say “ f ( x ) approaches infinity as x approaches infinity,” and we write f ( x )→∞ as x →∞. For example, for the function f ( x ) =3 x 2 , the outputs f ( x ) become larger as the inputs x get larger. We can conclude that the function f ( x ) =3 x 2 approaches infinity as x approaches infinity, and we write 3 x 2 →∞ as x →∞. The behavior as x →−∞ and the meaning of f ( x )→−∞ as x →∞ or x →−∞ can be defined similarly. We can describe what happens to the values of f ( x ) as x →∞ and as x →−∞ as the end behavior of the function. To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. The behavior for higher-degree polynomials can be analyzed similarly. Consider a quadratic function f ( x ) = ax 2 + bx + c . If a >0, the values f ( x )→∞ as x →±∞. If a <0, the values f ( x )→−∞ as x →±∞. Since the graph of a quadratic function is a parabola, the parabola opens upward if a >0; the parabola opens downward if a <0. (See Figure 1.19 (a).) Now consider a cubic function f ( x ) = ax 3 + bx 2 + cx + d . If a >0, then f ( x )→∞ as x →∞ and f ( x )→−∞ as x →−∞. If a <0, then f ( x )→−∞ as x →∞ and f ( x )→∞ as x →−∞. As we can see from both of these graphs, the leading term of the polynomial determines the end behavior. (See Figure 1.19 (b).)
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