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Functions
Introduction to Functions
Definition – A function f from a set A to a
set B is a relation that assigns to each
element x in the set A exactly one
element y in the set B. The set A is the
domain of the function f, and the set B
contains the range
Characteristics of a Function
1.
2.
3.
4.
Each element in A must be matched
with an element in B.
Some elements in B may not be
matched with any element in A.
Two or more elements in A may be
matched with the same element in B.
An element in A (domain) cannot be
match with two different elements in B
Example
A = {1,2,3,4,5,6} and B =
{9,10,12,13,15}
Is the set of ordered pairs a
function?
{(1,9), (2,13), (3,15), (4,15),
(5,12), (6,10)}
Vertical Line Test
Use the vertical line test to
determine graphically when you
have a function. If you can draw
a vertical line and it does not pass
through more than one point on
the graph, then the graph depicts
a function.
Function Notation
The variable f is usually used to depict a
function. It is only notation, and f(x) simply
replaces y in your typical equations and is
read f of x.
Therefore y = f(x)
That means if y = 2x +4 then an equivalent
equation using function notation is
f(x) = 2x + 4
Nothing changes, it’s just another use of
symbols.
Example
Evaluate the function when x = -1, 0, and 1
f(x) = { x2 +1, x< 0
{ x -1,
x≥0
f(-1) = (-1)2 +1 = 2
f(0) = 0 -1 = -1
f(1) = 1 – 1) = 0
f(x) = 1 – x2 then
f(1) = 1 – (1) 2 = 0
f(2) = 1 – (2) 2 = -3
f(0) = 1 – (0) 2 = 1
Domain of a Function
The domain of a function is the set of all real
numbers for which the expression is
defined.
EXAMPLE
f(x) = 1/(x2 -4)
The domain is the set of real numbers
excluding ± 2.
Analyzing Graphs of
Functions
Graph of a Function
The graph of a function f is the collection of
ordered pairs (x,f(x)) such that x is in the
domain of f.
Domain – is the set of all x values
Range – is the set of all f(x) values
Zeros of a Function
The zeros of a function f of x are the x-values
for which f(x) = 0
EXAMPLE
Find the zeros of f(x) = 3x2 +x - 10
3x2 +x – 10 = 0 - Factor and solve for x
Increasing and Decreasing
Functions
1.
A function f is increasing on an interval if,
for any x1 and x2 in the interval, x1 < x2
implies f(x1) < f(x2)
A function f is decreasing on an interval
if, for any x1 and x2 in the interval, x1 < x2
implies f(x1) > f(x2)
3. A function f is constant on an interval if,
for any x1 and x2 in the interval, f(x1) =
f(x2)
2.
Example
1.
Graph f(x) = x3
2.
Graph f(x) = x3 – 3x
3.
Graph f(x) = {x +1, x < 0
{1, 0 ≤ x ≤ 2
{ -x + 3, x >2
Definition of Relative Minimum and
Relative Maximum


A function value f(a) is called a relative
minimum of f if there exist an interval
(x1, x2) that contains a such that
x1 < x < x2 implies f(a) ≤ f(x)
A function value f(a) is called a relative
maximum of f if there exist an interval
(x1, x2) that contains a such that
x1 < x < x2 implies f(a) ≥ f(x)
Example
1.
Graph f(x) = 3x2 – 4x -2 using a calculator
to estimate the relative minimum or
relative maximum
2.
Graph f(x) = -3x2 + 4x + 2 using a
calculator to estimate the relative
minimum or relative maximum
Types of Functions

Linear Functions:
f(x) = mx + b

Step Functions:
f(x) = [[ x ]] = greatest integer less than or equal to x

Piecewise-Defined Functions:
f(x) = {2x +3, x ≤ 1
{- x + 4, x > 1
Even and Odd Functions

A function y = f(x) is even if, for each x in
the domain of f, f(-x) = f(x) --symmetric to y-axis

A function y = f(x) is odd if, for each x in
the domain of f, f(-x) = - f(x) --symmetric to origin
Example
Determine whether each function is even,
odd, or neither
1.
g(x) = x3 –x
2.
h(x) = x2 + 1
Shifting, Reflecting, and
Stretching Graphs
Summary of Graphs of
Common Functions






f(x)
f(x)
f(x)
f(x)
f(x)
f(x)
=
=
=
=
=
=
c
x
|x|
x
2
x
x3
Shifting Graphs

Transforms graphs by shifting
upward, downward, left or
right with basic graph the
same.
EXAMPLE
h(x) = x2 + 2 shifts the graph
upward two units
Vertical Shifts

h(x) = f(x) + c for c > 0
 Vertical shift c units upward

f(x) = f(x) – c for c > 0
 Vertical shift c units downward
Horizontal Shifts

h(x) = f(x – c) for c > 0
 horizontal shift c units right

f(x) = f(x + c) for c > 0
 horizontal shift c units left
Reflecting in the Coordinate Axes

Reflections in the x-axis:
 h(x) = - f(x)

Reflections in the y-axis:
 h(x) = f(-x)
Reflecting Graphs

Transforms graphs by
creating a mirror image
EXAMPLE
If h(x) = x2 then g(x) = - x2 is
the reflection
Nonrigid Transformation
Transformations that cause a
distortion – a change in the
shape of the original graph
If h(x) = |x| then g(x) = 3|x| is a
vertical stretch of h(x)
but p(x) = ⅓|x| would be a vertical
shrink
Combinations of Functions
Arithmetic Combinations of
Functions

(f +g)(x) = f(x) + g(x) sum

(f -g)(x) = f(x) - g(x) difference

(fg)(x) = f(x) · g(x) product

(f/g)(x) = f(x)/g(x), g(x) ≠ 0 quotient
Examples
f(x) = 2x + 1 and g(x) = x2 + 2x – 1
Find: (f +g)(x)
Find: (fg)(x)
= f(x) + g(x)
= x2 + 4x
= f(x)· g(x)
=2 x3 +5x2 - 1
Composition of Functions
The composition of the function f with the
function g is (f ◦ g)(x) = f(g(x))
The domain of f ◦ g is the set of all x in
the domain of g such that g(x) is in the
domain of f
Examples
f(x) = x + 2 and g(x) = 4 – x2
Find: (f ◦ g)(x)
= f(g(x))
= f(4 – x2)
Simplify
= 4 – x2+2
Find: (g ◦ f)(x)
Simplify
= g(f(x))
= g(x + 2)
= 4 – (x +2)2
Examples
f(x) = x2 - 9 and g(x) = (9 - x2)½
Find: domain of (f ◦ g)
Remember the domain of (f ◦ g) is the
set of all x in the domain of g
Find domain of g(x):
Examples
How to decompose a composite function
1.
2.
Look for an inner function, then
Look for n outer function
Example: h(x) = (3x – 5)3
Let g(x) = (3x-5), then f(x) = x3
h(x) = (f ◦ g)(x)
Inverse of Functions
Inverse Functions
Let f and g be two functions such that
f(g(x)) = x for every x in the domain
of g and;
g(f(x)) = x for every x in the domain of f
Under these conditions, the function g is
the inverse function of the function f
Inverse Functions
The inverse function is formed by
interchanging the first and second
coordinates of each of the ordered pairs
and the inverse is denoted by f -1
Again, this is simply notation!
The domain of f must be equal to the
range of f -1 , and the range of f must
be equal to the domain of f-1
Example
Find the inverse function of
f(x) = 2x - 3
Replace f(x) with y and solve for x
y = 2x -3
x = (y+3)/2
Now interchange x and y and you
have f -1
y = (x+3)/2
Guidelines for Finding an
Inverse Function
1.
Use the Horizontal Line Test to decide
whether f has an inverse function
2.
In the equation for f(x), replace f(x) by y
3.
Interchange the roles of x and y, and solve
for y.
4.
Replace y by f-1(x) in the new equation
5.
Verify that f and f-1 are inverse functions of
each other by showing that the domain of f
is equal to the range of f -1 and the range of
f is equal to the domain of f -1
Mathematical Modeling
Direct Variation
The following statements are equivalent.
1.
2.
3.
y varies directly as x.
y is directly proportional to x
y = kx for some nonzero constant k
EXAMPLE
D = rt
F= ma
Inverse Variation
The following statements are equivalent.
1.
2.
3.
y varies inversely as x.
y is inversely proportional to x
y = k/x for some nonzero constant k
EXAMPLE
V = kT/P
END