Download Exponential and Logarithmic Functions

Exponential and Logarithmic
Functions
Composite Functions
Inverse Functions
Exponential Function Intro
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’
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.
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.
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
f(x) = 1/(x+2) g(x) = 4/(x-1)






Find
Find
Find
Find
Find
the domain of the composite f o g
fog
the domain of the composite g o f
g of
(g o f)(4)
Find the domain of f o g if f(x) = square
root of x and g(x) = 2x + 3
Find the components of the
following composites:

H(x) = (x2 + 1)50

S(x) = 1 / (x + 1)
Show that the two composite
functions are equal for:

f(x) = 3x – 4

f

gof=

Look at number 8 on page 397
o
g(x) = (1/3)(x + 4)
g=
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
Determine if the following
functions are inverses

f(x) = x3

f(x) = 3x + 4
g(x) = cube root of x
f-1(x) = (1/3)(x – 4)
Finding inverses



Ordered Pairs: reverse the x and y
Equations: reverse x and y then solve
for y
Graphs: Invert x’s and y’s off of original
graph, plot new points
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
Laws of Exponents

as . at = as+t

(as)t = ast

(ab)s = as . bs

1s = 1

a0 = 1

a-s = 1/as
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?
Transformations: work same as
on quadratic

F(x) = 3-x + 2





Up 2, reflect across x-axis
Horizontal asymptote at y=2
F(x) = 2x-3 – 5
Right 3, down 5
Horizontal asymptote at y=-5
Examples

Page 423, #15, 23, 31, 34, 44, 74
Solving an Exponential Equation



If au = av, then u = v
Get bases equal, then set exponents
equal and solve.
3x+1 = 81
More examples

Page 425; #54, 58, 62, 68, 66
Base e

E = (1 + 1/n)n
infinity
as n approaches

Look at Page 419 – bottom of page

Approximate value?

Called the natural base
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
Application Examples
Page 426 #80, 88
Assignment

Page 397, 409, 423