Chapter 3 Exponential and Logarithmic Functions Section 3

Chapter 3 Exponential and Logarithmic Functions

Section 3 -1 Exponential Functions and Their Graphs

Pensamiento n "La verdadera felicidad se logra en el esfuerzo, no en la diversión. “ Bonifacio

Exponential Functions

We have dealt with algebraic functions, which included polynomial and rational functions

We will study two types of nonalgebraic functions: exponential and logarithmic functions.

Definition: The exponential function f with base a is denoted by f(x) = ax where a > 0 , a ≠ 1 and x is any real number

Graphs of Exponential Functions

Note n Graphs of exponential functions have similar characteristics

Example 1. In the same coordinate plane, sketch the graph of each function a. f(x) = 2 x b. g(x) = 4 x

f(x) = 4 x f(x) = 2 x

Example 2. In the same coordinate plane, sketch the graph of each function a. F(x) = 2 -x b. G(x) = 4 -x

F(x) = 2 -x G(x) = 4 -x

Note - Observe F(x) = 2 –x = f(-x) and G(x) = 4 –x = g(-x) - F is a reflection of f (in y-axis)

Note - The graphs are typical of the exponential functions a x and a –x - They have one y-intercept and one horizontal asymptote (x-axis) - They are continuous

Basic Characteristics Graph of y = ax a>0, a≠ 1 Graph of y = a-x a>0, a≠ 1 - Domain (-∞, ∞) - Range (0 , ∞) - y-intercept (0 , 1) - Increasing - Decreasing - x-axis HA - Continuous

Excersis n a. b. c. d. Graph each transformation of the graph f(x) = 3 x in the same coordinate plane g(x) = 3 x + 1 h(x) = 3 x – 2 k(x) = - (3 x ) j(x) = 3 –x

The natural base e

Notes - e ≈ 2. 71828… (irational) - This number is called the natural base - The function f(x) = e x is called the natural exponential function

Example Sketch the graph of each natural exponential a. f(x) = 2 e 0. 24 x b. g(x) = ½ e -. 058 x