Exponential Functions

Definition and Basic Properties

An exponential function is a function of the form , where a is a positive real number (the base), and x is any real number. The domain of an exponential function is all real numbers, and the range is all positive real numbers.

• Growth Factor: The base a determines whether the function is increasing or decreasing.

• Initial Value: for any base .

• One-to-One: Exponential functions are one-to-one, meaning each input yields a unique output.

• Horizontal Asymptote: The x-axis () is a horizontal asymptote.

Example: is an exponential function with base 3.

Example: is an exponential function with base less than 1.

Laws of Exponents

The laws of exponents are fundamental rules for manipulating exponential expressions:

Identifying Linear vs. Exponential Functions

To distinguish between linear and exponential functions, examine the average rate of change and the ratio of consecutive outputs:

• Linear Function: Constant average rate of change (slope).

• Exponential Function: Constant ratio of consecutive outputs.

Example Table: Linear Function

Example Table: Exponential Function

Graphing Exponential Functions

Exponential functions can be graphed by plotting points and observing their behavior:

• For , the function increases rapidly as increases.

• For , the function decreases rapidly as increases.

• The y-intercept is always 1.

• The x-axis is a horizontal asymptote.

General Graphs:

Transformations of Exponential Functions

Exponential functions can be transformed by reflecting, shifting, or stretching:

• Reflection: reflects about the y-axis.

• Vertical Shift: shifts the graph up by units.

• Horizontal Shift: shifts the graph right by units.

Example: Transformations of and

The Number e

The number e is a fundamental mathematical constant, approximately equal to 2.71828. It is defined as the limit:

Exponential functions with base are called natural exponential functions.

Example: Graph of

Graphing Exponential Functions Using Transformations (with e)

Transformations can also be applied to functions with base :

• Reflections, shifts, and stretches follow the same rules as for other bases.

Solving Exponential Equations

To solve exponential equations, rewrite each side with the same base and use properties of exponents:

• If , then .

• Use logarithms if the bases cannot be made equal.

Example: Solve . Since , .

Applications: Exponential Probability

Exponential functions are used in probability and statistics to model events such as arrival times:

• The probability that an event occurs within minutes is , where is the rate parameter.

Example: Probability that a car arrives within 5 minutes:

Example: Probability that a car arrives within 30 minutes:

Graph: Probability function as increases:

Summary Table: Properties of Exponential Functions