## Exponential Functions

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Exponentials: Basics :: Exponent Rules :: Graphing :: Practice Problems :: Quiz

**BASICS:** (top)

An **exponential function** has the form for b>0, where a is the** initial value** and b is the** base**.

There are two main types of exponential functions: **exponential growth** and **exponential decay**, which are pictured below. An exponential function of the form is an exponential growth function when b>1 and an exponential decay function when 0<b<1. Note that if b≤0, the function is not an exponential function.

**EXPONENT RULES:** (top)

It can be very useful to rewrite exponential functions in different ways. This section outlines the rules that make such rearragements possible.

__Definitions__

- (b multiplied by itself n times)

__Other Rules__

1. Multiplication Rule:

2. Divison Rule:

3. Power Rule:

4. Multiplicative Distribution:

5. Quotient Distribution:

Note that it is NOT necessarily the case (indeed, almost always is NOT the case) that or .

EXAMPLE 1:

Problem:Simplify the following exponential expression using the definitions and rules from above: .

Solution:First, let's use the quotient rule to simplify inside each set of parenthesis. This gives us . Now, change the division to multiplication using the invert and multiply rule to get . Next, use the power rule to get rid of the exponents outside the parenthesis: . Seeing the -10n^{2}in both the numerator and denominator makes us want to cancel it out. To do so, we must use the multiplication rule to separate the exponents in the numerator: . Then we can cancel to get the final answer of .

EXAMPLE 2:

Problem:Simplify the following exponential expression using the definitions and rules from above:

Solution:First, get rid of the negative exponents to get . Next, convert the fractional exponenents into their root and power forms: . Evaluate these denominators to find , then evaluate the expression to get the final answer of .

**GRAPHING:** (top)

The domain of any exponential function is . Note that , so the function's y-intercept is at y=a. The range of an exponential is either all y>0, if a is positive, or all y<0, if a is negative. Note that y=0 is not in the range; thus, all exponential functions have a **horizontal asymptote **at y=0 and do not have any x-intercepts.

EXAMPLE 3:

Problem:Graph the function .

Solution:The graph is exponential decay because the base is between 0 and 1. It has a positive initial value, so the graph is always positive and has a horizontal asymptote at y=0. It looks like this:

**PRACTICE PROBLEMS:** (answers | solutions | top)

- Simplify the exponential expression . (answer | solution)
- Simplify the exponential expression . (answer | solution)
- Graph the function . (answer | solution)

**QUIZ:** (top)

This page last updated 22 June, 2008 5:32 PM