Exponential and Logarithmic Functions

1
Exponential and Logarithmic Functions

• Composite Functions
• Inverse Functions
• Exponential Function Intro

2
Objectives

• Form a composite function and find its domain
• Determine the inverse of a function
• Obtain the graph of the inverse from the graph of
  a function
• Evaluate and graph an exponential function
• Solve exponential equations
• Define the number e

3
Composite Functions

• Combining of two or more processes into one
  function
• (f o g)(x) (f(g(x))) read as f composed with
  g
• The domain is the set of all numbers x in the
  domain of g such that g(x) is in the domain of f.

4
Look at diagrams on page 392 of text book.

• In figure 1, the top value of x would not be in
  the composite domain since the range of g does
  not exist in the domain of f.

5
Examples

• Suppose f(x) 2x and g(x) 3x2 1
• Find (f o g)(4)
• Find (g o f)(2)
• Find (f o f)(1)
• Find (f o g)(x)
• Find (g o f)(x)
• Find the domain of the composite

6
f(x) 1/(x2) g(x) 4/(x-1)

• Find the domain of the composite f o g
• Find f o g
• Find the domain of the composite g o f
• Find g o f
• Find (g o f)(4)
• Find the domain of f o g if f(x) square root of
  x and g(x) 2x 3

7
Find the components of the following composites

• H(x) (x2 1)50
• S(x) 1 / (x 1)

8
Show that the two composite functions are equal
for

• f(x) 3x 4 g(x) (1/3)(x 4)
• f o g
• g o f
• Look at number 8 on page 397

9
When both composites end up with x as the final
range they are inverse functions.

• Inverse functions when a function manipulates
  the range of one function and outputs the
  original domain
• To Test Each of the following must be true
• (f o g)(x) x
• (g o f)(x) x

10
Determine if the following functions are inverses

• f(x) x3 g(x) cube root of x
• f(x) 3x 4 f-1(x) (1/3)(x 4)

11
Finding inverses

• Ordered Pairs reverse the x and y
• Equations reverse x and y then solve for y
• Graphs Invert xs and ys off of original
  graph, plot new points

12
Exponential Functions

• f(x) ax
• a is a positive real number a ? 0, domain is the
  set of all real numbers
• a is called the base number
• x is called the exponent

13
Laws of Exponents

• as . at ast
• (as)t ast
• (ab)s as . bs
• 1s 1
• a0 1
• a-s 1/as

14
Graphs of Exponential Functions

• f(x) (1/2)x f(x) 2x
• Plug numbers in for x and graph
• Look at function values at f(1)
• Look at bases what happens when base is
  fraction? When base is whole value?
• As base gets bigger what happens to graph?

15
Transformations work same as on quadratic

• F(x) 3-x 2
• Up 2, reflect across x-axis
• Horizontal asymptote at y2
• F(x) 2x-3 5
• Right 3, down 5
• Horizontal asymptote at y-5

16
Examples

• Page 423, 15, 23, 31, 34, 44, 74

17
Solving an Exponential Equation

• If au av, then u v
• Get bases equal, then set exponents equal and
  solve.
• 3x1 81

18
More examples

• Page 425 54, 58, 62, 68, 66

19
Base e

• E (1 1/n)n as n approaches infinity
• Look at Page 419 bottom of page
• Approximate value?
• Called the natural base

20
Graph

• F(x) ex
• F(x) -ex-3
• Look at translations
• Same as translations for other functions
• Add/Subtract after base vertical shift
• Add/Subtract in process horizontal shift
• Negative reflection
• Numbers multiplied Stretch/Compression

21
Application Examples

• Page 426 80, 88

22
Assignment

• Page 397, 409, 423