2.1 Functions and Their Graphs
By
Dr. Julia Arnold
A function is a rule
that assigns to each
element in a set A
one and only one
element in a set B.
Domain-All
elements in
domain are
assigned to
something
Range- The range
consists of the
elements used by
the domain.
The Range
In general our domain will begin with the Real Numbers.
However, there are some equations which require us to use a
subset of the Reals for the domain.
These equations are:
1. Certain types of word problems which pertain to measurable
items. For example Volume of a box in terms of the size of
material.
2.Equations where the variable is in the denominator of a
fraction: y  2
x 5
Equations which contain the variable under a radical:
Or a combination of the above.
y  x 7
y=x
y = x3
y
y
3
2
1
-4
-3
-2
-1
1
2
-1
-2
3
4
x
5
In pre-calculus
you studied
the graphs of
some common
functions
3
2
1
-4
-4
y
2
1
-2
-1
1
-1
-2
Continued
-3
-4
2
1
-3
3
-3
-1
-2
y = x2
-4
-2
-1
-3
-4
-3
3
4
x
5
2
3
4
x
5
Functions continued:
y= x
y =x
y
3
2
1
y
3
-4
-3
-2
-1
1
2
3
x
4
5
-1
2
-2
1
-3
-4
-4
-3
-2
-1
1
-1
-2
1
y= x
-3
-4
y
3
2
1
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
x
5
2
3
4
x
5
The Vertical-Line Test shows you which graphs are
functions: If you pass a vertical line across the graph
it should only intersect the graph one point at a time.
Non-functions:
y
y
y
3
2
1
-4
-3
-2
-1
1
2
3
4
x
5
-4
-3
-2
3
3
2
2
1
1
-1
1
2
3
4
x
5
-4
-3
-2
-1
1
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-1
2
3
4
x
5
Problem
Find the domain of the function:
f(x) 
x 1
(x  2)(x  3)
This problem has both a radical and a fraction.
We must find the numbers which keep the radicand
positive and the denominator non-zero.
Solution: x  2, x  3 or the denominator would be zero.
In order that the radicand is positive or zero
x 1  0
x 1
-1
0
-2
1
2
3
Since the domain must be greater than or equal to one,
we don’t have to be concerned with -2. However, 3 is
greater than 1 but must not be in the domain.
So the domain is 1,3 3, 
Sketch the graph of the function with the given rule.
Find the domain and range of the function.
 x  1, ifx  1

f(x)   0, if  1  x  1
 x  1, ifx  1

This is called a piece-wise
function. It has 3 pieces.
The domain is represented by the 3 if statements:
x < -1  1  x  1 ,x > 1 which when put together is
all real numbers.
The first and last equation will be slanted lines.
The middle equation is a horizontal line.
Problem 46
Sketch the graph of
the function with the
given rule. Find the
domain and range of the
function.
y
 x  1, ifx  1

f(x)   0, if  1  x  1
 x  1, ifx  1

The range is
the set of numbers
used in the graph for
the y value.
y0
3
2
1
-4
-3
-2
-1
1
-1
-2
-3
-4
2
3
4
x
5