# 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) = 3^{x}

**2. Graphing Exponential Functions**

An exponential function is a function of the form

f(x) = a^{x}

where a is a positive number and a ≠ 1.

Example:

Graph f(x) = 2^{x}.

**2. Properties of Exponential Functions**

** Properties of the Graph of an Exponential
Function f(x) = a ^{x}, 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) = a ^{x}, 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 a^{x}, 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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