Exponential Functions Section 11.3 Exponential Functions

Objectives Define exponential functions Graph exponential functions Use exponential functions in applications involving growth or decay

Objective 1: Define Exponential Functions An exponential function with base b is defined by the equations ƒ(x) = bx or y = bx where b > 0, b ≠ 1, and x is a real number. The domain of ƒ(x) = bx is the interval (–∞, ∞) and the range is the interval (0, ∞).

Objective 2: Graph Exponential Functions Since the domain and range of ƒ(x) = bx are sets of real numbers, we can graph exponential functions on a rectangular coordinate system. Properties of Exponential Functions: The domain of the exponential function ƒ(x) = bx is the interval (–∞, ∞) and the range is the interval (0, ∞). The graph has a y-intercept of (0, 1). The x-axis is an asymptote of the graph. The graph of ƒ(x) = bx passes through the point (1, b). Exponential functions are one-to-one.

Objective 2: Graph Exponential Functions 6. If b > 1 then ƒ(x) = bx is an increasing function. If 0 < b < 1, then ƒ(x) = bx is a decreasing function.

EXAMPLE 1 Graph ƒ(x) = 2x Strategy We will graph the function by creating a table of function values and plotting the corresponding ordered pairs. Why After drawing a smooth curve through the plotted points, we will have the graph.

EXAMPLE 1 Solution Graph ƒ(x) = 2x To graph ƒ(x) = 2x, we choose several values for x and find the corresponding values of ƒ(x). If x is –3, and if x is –2, we have: The point and are on the graph of ƒ(x) = 2x. In a similar way, we find the corresponding values of ƒ(x) for x values of –1, 0, 1, 2, 3, and 4 and list them in a table. Then we plot the ordered pairs and draw a smooth curve through them.

EXAMPLE 1 Graph ƒ(x) = 2x Solution From the graph, we can see that the domain of ƒ(x) = 2x is the interval (–∞, ∞) and the range is the interval (0, ∞). Since the graph passes the horizontal line test, the function is one-to-one. Note that as x decreases, the values of ƒ(x) decrease and approach 0. Thus, the x-axis is a horizontal asymptote of the graph. The graph does not have an x-intercept, the y–intercept is (0, 1), and the graph passes through the point (1, 2).

Objective 3: Use Exponential Functions in Applications Involving Growth or Decay Formula for Compound Interest: If $P is deposited in an account and interest is paid k times a year at an annual rate r, the amount A in the account after t years is given by

Professional Baseball Salaries Professional Baseball Salaries. The exponential function s(t) = 650,000(1.09)t approximates the average salary of a major League baseball player, where t is the number of years after 1990. (Source: Baseball Almanac) a. Graph the function. b. Use the function to determine the average annual salary in 2020, if the current trend continues. EXAMPLE 4 Strategy For part a, we will graph the function by creating a table of function values and plotting the resulting ordered pairs. For part b, we will find s(30). Why After drawing a smooth curve though the plotted points, we will have the graph. Since the year 2020 is 30 years after 1990, t = 30.

Professional Baseball Salaries Professional Baseball Salaries. The exponential function s(t) = 650,000(1.09)t approximates the average salary of a major League baseball player, where t is the number of years after 1990. (Source: Baseball Almanac) a. Graph the function. b. Use the function to determine the average annual salary in 2020, if the current trend continues. EXAMPLE 4 Solution a. The function values for t = 0 and t = 5 are computed as follows: Using a scientific calculator, we find the corresponding values of s(t) for t-values of 10, 15, 20, and 25 and list them in a table :

Professional Baseball Salaries Professional Baseball Salaries. The exponential function s(t) = 650,000(1.09)t approximates the average salary of a major League baseball player, where t is the number of years after 1990. (Source: Baseball Almanac) a. Graph the function. b. Use the function to determine the average annual salary in 2020, if the current trend continues. EXAMPLE 4 Solution Then we plot the ordered pairs and draw a smooth curve through them to get the graph.

Professional Baseball Salaries Professional Baseball Salaries. The exponential function s(t) = 650,000(1.09)t approximates the average salary of a major League baseball player, where t is the number of years after 1990. (Source: Baseball Almanac) a. Graph the function. b. Use the function to determine the average annual salary in 2020, if the current trend continues. EXAMPLE 4 Solution b. To estimate the average annual major league baseball salary in 2020, which is 30 years after 1990, we find s(30). If the trend continues, in 2020, the average annual salary will be approximately $8,623,991.