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Exponential Functions. The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than and x is any real number. /. Base is 2. Base is 10. Base is 3. Definition of the Exponential Function.

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definition of the exponential function

The exponential function f with base b is defined by

f (x) = bx or y = bx

Where b is a positive constant other than and x is any real number.

/

Base is 2.

Base is 10.

Base is 3.

Definition of the Exponential Function

Here are some examples of exponential functions.

f (x) = 2xg(x) = 10xh(x) = 3x+1

text example

f (x) = 13.49(0.967)x – 1

This is the given function.

f (31) = 13.49(0.967)31 – 1

Substitute 31 for x.

Press .967 ^ 31 on a graphing calculator to get .353362693426. Multiply this by 13.49 and subtract 1 to obtain

f (31) = 13.49(0.967)31 – 1=3.77

The exponential function f (x) = 13.49(0.967)x – 1 describes the number of O-rings expected to fail, f (x), when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, colder than any previous experience. Find the number of O-rings expected to fail at this temperature.

Text Example

Solution Because the temperature was 31°F, substitute 31 for x and evaluate the function at 31.

characteristics of exponential functions

The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.

  • The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.
  • If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function.
  • If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function.
  • f (x) = bx is a one-to-one function and has an inverse that is a function.
  • The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote.

Characteristics of Exponential Functions

f (x) = bx

0 < b < 1

f (x) = bx

b > 1

transformations involving exponential functions

Transformation

Equation

Description

Horizontal translation

g(x) = bx+c

  • Shifts the graph of f (x) = bx to the left c units if c > 0.
  • Shifts the graph of f (x) = bx to the right c units if c < 0.

Vertical stretching or shrinking

g(x) = c bx

  • Multiplying y-coordintates of f (x) = bx by c,
  • Stretches the graph of f (x) = bx if c > 1.
  • Shrinks the graph of f (x) = bx if 0 < c < 1.

Reflecting

g(x) = -bx

g(x) = b-x

  • Reflects the graph of f (x) = bx about the x-axis.
  • Reflects the graph of f (x) = bx about the y-axis.

Vertical translation

g(x) = -bx + c

  • Shifts the graph of f (x) = bx upward c units if c > 0.
  • Shifts the graph of f (x) = bxdownward c units if c < 0.

Transformations Involving Exponential Functions

text example1

x

f (x) = 3x

g(x) = 3x+1

g(x) = 3x+1

f (x) = 3x

-2

3-2 = 1/9

3-2+1 = 3-1 = 1/3

-1

3-1 = 1/3

3-1+1 = 30 = 1

0

30 = 1

30+1 = 31 = 3

1

31 = 3

31+1 = 32 = 9

(-1, 1)

(0, 1)

2

32 = 9

32+1 = 33 = 27

1

2

3

4

5

6

-5

-4

-3

-2

-1

Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.

Text Example

Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs.

the natural base e

f (x) = 3x

f (x) = ex

4

(1, 3)

f (x) = 2x

3

(1, e)

2

(1, 2)

(0, 1)

1

An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately,

The number e is called the natural base. The function f (x) = ex is called the natural exponential function.

The Natural Base e

-1

formulas for compound interest
Formulas for Compound Interest
  • After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:
  • For n compoundings per year: A = P(1 + r/n)nt
  • For continuous compounding: A = Pert.
example
Use A= Pert to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously

Solution:

A= PertA = 2000e(.07)(8)A = 2000 e(.56)A = 2000 * 1.75A = $3500

Example