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

  1. (b multiplied by itself n times)

Other Rules

1. Multiplication Rule:

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2. Divison Rule:

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3. Power Rule:

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4. Multiplicative Distribution:

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5. Quotient Distribution:

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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 -10n2 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)

  1. Simplify the exponential expression . (answer | solution)
  2. Simplify the exponential expression . (answer | solution)
  3. Graph the function . (answer | solution)

QUIZ: (top)

Exponentials (1 questions)
Simplifying exponentials.

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Then, click Start Test.

 

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