# Exponential Functions

Learning Objectives:

1. Evaluate exponential expressions .

2. Graph exponential functions.

3. Define the number e .

4. Solve exponential equations .

1. Evaluate Exponential Functions

Example: f(x) = 3x

2. Graphing Exponential Functions

An exponential function is a function of the form

f(x) = ax

where a is a positive number and a ≠ 1.

Example:
Graph f(x) = 2x.

2. Properties of Exponential Functions

Properties of the Graph of an Exponential
Function f(x) = ax, a > 1

1. The domain is the set of all real numbers . The
range is the set of all positive real numbers.

2. There are no x-intercepts; the y- intercept is 1.

3. The graph of f contains the points
(0,1), and (1, a).

2. Graphing Exponential Functions

Example:

Graph

2. Properties of Exponential Functions

Properties of the Graph of an Exponential
Function f(x) = ax, 0 < a < 1

1. The domain is the set of all real numbers . The
range is the set of all positive real numbers.

2. There are no x-intercepts; the y- intercept is 1.

3. The graph of f contains the points
(0,1), and (1, a).

3. The Number e

The number e is defined as the number that the
expression

approaches as n becomes unbounded in the positive
direction (that is, as n gets bigger).

e ≈ 2.718

4. Solving Exponential Equations-Basic

Equations that involve terms of the form ax, a > 0, a ≠ 1
are called exponential equations .

Property for Solving Exponential Equations

4. Solving Exponential Equations

4. Exponential Models

Example:
Newton’s Law of Cooling states that the temperature of a heated
object decreases exponentially over time toward the temperature of
the surrounding medium. Suppose that a pizza is removed from a
400° oven and placed in a room whose temperature is 70°. The
temperature u (in °F) of the pizza at time t (in minutes) can be
modeled by What will be the temperature of the
pizza after 10 minutes?

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