1.1 - Functions
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Interval Notation
Graph
Algebraic Notation
Interval Notation
|
3
x>3
-1 ≤ x < 5
|
-1
|
5
x < -2 or x ≥ 4
|
-2
|
4
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Function, Domain, and Range
A function is a relationship or correspondence between
two sets of numbers, in which each member of the first
set (called the domain) corresponds to one an only one
member of the second set (called the range).
3
f
x1
y1
x2
y2
x3
y3
X
Domain
Y
Range
4
Functions? State the domain and range.
-2
1
-4
1
3
-2
1
-8
5
7
6
-2
-2
5
7
5
Explanations
If asked, how do you state that a relation is or is not a
function?
Yes, it is a function. Each domain element corresponds
to exactly one range element.
No, it is not a function. Some domain element
corresponds to more than one range value.
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Function Representation
1.
2.
3.
4.
Verbal or Written
Numerically
Graphically
Symbolically
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Examples
1. A correspondence between the students in this class
and their student identification numbers.
Do the following relations represent functions?
2.  2,3 1,3  2,5  10,5 
3. 1, 2   2, 2   3,1  4, 2 
4.  0, 0  1, 0   3, 0   5, 0 
8
y
Function?
x
9
y
x
10
Theorem Vertical Line Test
If any vertical line drawn on the graph of a
relation crosses the graph more than once, the
relation does not represent a function.
Caution: It is not sufficient to state
that a graph represents a function
because it passes the vertical line
test.
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Determine the domain, range, and intercepts of the
following graph.
y
4
0
-4
(2, 3)
(1, 0)
(4, 0)
(10, 0)
x
(0, -3)
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Example Does this equation represent a function?
Why or why not?
(x – 2)2 + (y + 4)2 = 25
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Linear Functions
y = mx + b [Slope-Intercept Form of a Line]
m = slope
b = y-intercept
y – y1 = m(x – x1) – [Point-Slope Form of a Line]
m = slope
(x1, y1) = point on line
y = y1 [Horizontal Line]
m=0
x = x1 [Vertical Line]
m = undefined or none
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Linear Functions - Examples
Determine the equation of the line through (-2, ¾) with
slope m = ½ .
Determine the equation of the line through (4, -1) with
no slope.
Determine the equation of the line through (-2, 5) with
m = 0.
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Example
Let y = 2x – 3
y = x2 + 2x – 1
y=|x+5|
If x = 3, then y = _____? There is only one
answer to this question.
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Example
Let f = 2x – 3
g = x2 + 2x – 1
h=|x+5|
If x = 3, then h = _____? There is only one
answer to this question.
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Function Notation
Read: “f of x”
f is the name of
the function
Does Not
Mean f times
x.
f (x)
x is the variable into
which we substitute
values or other
expressions
x is called the independent variable
and f(x) is the dependent variable.
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Find the domain of the following functions:
(a)
f ( x)  2 x  1
(b)
x
g ( x) 
x 1
(c)
h( x)  4  x
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Evaluating Functions
Let f(x)=2x – 3 and g(m) = | m2 – 2m + 1|.
Determine:
(a) f(-3)
(b) g(-2)
(c) f(a)
(d) g(a + 1)
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The Difference Quotient
f ( x  h)  f ( x )
h
Example
If f (x) = x2 – 3, determine the difference quotient.
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The Difference Quotient
First, we need to determine f (x + h)
f (x + h) = ( x +xh )2 – 3
= (x2 + 2hx + h2) – 3
= x2 + 2hx + h2 – 3
f (x) = x2 – 3
 f ( x  h)   f ( x )
h
[ x 2  2hx  h 2  3]  [ x 2  3]
h
2hx  h 2 h(2 x  h)

 2x  h
h
h
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The Difference Quotient
Example
Determine the difference quotient for each of
the following and simplify.
1. f  x   x  2
x
2. g  x  
x 5
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Increasing and Decreasing Functions



A function f is increasing on an open interval I if, for
any choice of x1 and x2 in I, with x1 < x2 we have f(x1)
< f(x2).
A function f is decreasing on an open interval I if, for
any choice of x1 and x2 in I, with x1 < x2 we have f(x1)
> f(x2).
A function f is constant on an open interval I if, for
all choices of x, the values f(x) are equal.
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y
4
Increasing, Decreasing Constant, Local
Maximum, Local Minimum
(2, 3)
(4, 0)
0
(1, 0)
x
(10, -3)
-4
(0, -3)
(7, -3)
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Piece Wise Defined Functions
When functions are defined by more
than one equation, they are called
piece-wise defined functions.
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Example
For the following piece-wise defined function:
 x  3 if  2  x  1

f ( x)  3
if x  1
 x  3 if x  1

a) Find f (-1), f (1), f (3).
b) Sketch a graph of f.
c) Find the domain of f.
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Absolute Value
 a if a  0
a 
 a if a  0
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Example
Use the definition to do the following:
(a) Explain why | -2 | = 2.
(b) Determine the exact value of | 3 – π |.
(c) Write | x – 2 | as a piece wise defined
function without absolute value bars.
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Modeling With Functions
Example 1 Express the surface area of a cube as a
function of its volume.
Example 2 A Norman window has the shape of a
rectangle surmounted by a semicircle. If the
perimeter of the window is 30 ft, express the area
A of the window as a function of the width x of the
window.
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Odd or Even Function
A function is odd if f (-x) = -f (x). Odd functions
exhibit origin symmetry on their graphs. This
means if you turn the graph upside down, it will
look the same.
A function is even if f (-x) = f (x). The graphs of
even functions will be symmetric to the y-axis.
Simply substitute -x for each x in the function and
determine if you get f (x) or -f (x). If you get
neither, it is neither odd nor even.
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Odd or Even Function
Determine if f x   x 2  2 is odd or even.
x 2
Determine if g x   3
is odd or even.
x  5x
2
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