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Lecture 4

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Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

2 exponent. It should not be confused with the power function t▯x▯ ▯ x , in which the vari- able is the base. N In Appendix G we present an alternativeIn general, an exponential function is a function of the form approach to the exponential and logarithmic functions using integral calculus. f▯x▯ ▯ ax where a is a positive constant. Let’s recall what this means. MATH140 - Lecture 4 Notesve integer, then a ▯ a ▯ a ▯ ▯▯▯ ▯ a Exponential Functions: n factors Exponential function is dependent on the increasing or decreasing function. In general,a positive integer, then it is of the form f(x) = a^x where a is a positive constant. 1 a ▯n▯ n a To understand exponential functions, first look at the graph of an exponential function. If x is a rational number, x ▯ p▯q, where p and q are integers and q ▯ 0, then y x p▯q q p q p a ▯ a ▯ s a ▯ (s a) x But what is the meaning of a if x is an irrational number? For instance, what is meant by 2s3 or 5 ? x To help us answer this question we first look at the graph of the function y ▯ 2 , where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the 1 domain of y ▯ 2 to include both rational and irrational numbers. There are holes in the graph in Figure 1 corresponding to irrational values of x.We want 0 x x 1 to fill in the holes by defining f▯x▯ ▯ 2 , where x ▯ ▯, so that f is an increasing function. In particular, since the irrational numse3 satisfies FIGURE 1 Based on an understanding of the graph, we can use the concept of 1:1 to understand ▯ 1.8 whether a function is valid. What we mean by this 1:1 is that no two points go to the same x value. Definition: A one to one function is a function where every x has at most one y. No two values of x give the same f(x) and no two values of y come from the same x. Formally, f(x) is one to one if: x1 ≠ x2 㲗 f^-1(y) = x Now that we know exponential functions are valid because they pass the 1:1 test, we can examine their properties. Below is a graph of members of the family of functions y = a^x where there are varying values of the base a. All the graphs pass through the same point (0, 1) because a^0 = 1 for a ≠ 0. There are three essential points to exponential functions: • If 0 < a < 1, then the exponential function decreases If a = 1, it is a constant • • If a > 1, it increases Note, as the base a gets larger, the exponential function grows more rapidly (for x • > 0) Similary ▯ a about the y-axis. ia ▯ 0) where x is any irrational number. Figure 2 1 shows how all the holes in Figure 1 have been filled to complete the graph of the function y f▯x▯ ▯ 2 , x ▯ ▯ y y (00▯1) x The graphs of members of the family of funcy ▯ axare shown in Figure 3 for var- 1 ious valu0s of the base a. Notice that all of these graphs pass through the same point ▯0, 1▯ FIGURE 2 because a ▯ 1 fora ▯ 0 . Notice also that as the base a gets larger, the exponential func- y=2®, x real (0,▯1) tion grows more rapidly (fx ▯10). (0,▯1) 0 x 1 1 y 0 x 2▯▯▯’ ”4▯▯’ 10® 4® 2® 1.5® 0 x 0 x 0 x (a) y=a®, ▯01 (a) y=a®, ▯01 x FIGURE 4 NIf ▯ FIGURE 4enapprxacheasx becomes large. , thenapproaches as x decreases through negative values. In botOne reason for the importance of the exponential1®unction lies in the following proper- cases the x-axis is a horizontal asymptote.ties. If x and y are rational numbers, then these laws are well known from elementaryper- matters are discussed in Section 2.6.
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