Math 1650 Lecture Notes
Jason Snyder, PhD.
§2.2
Graphs of Functions
§ 2.2: Graphs of Functions
Graphing Functions
The Graph of a Function
If f is a function with domain A, then the graph of f is the set of ordered pairs
{ 𝑥, 𝑓 𝑥 : 𝑥 ∈ 𝐴}
In other words, the graph of f is the set of all points (x,y) such that y = f(x); that is, the
graph of f is the graph of the equation y = f(x).
Example 1 Graphing Functions
Sketch the graph of the following functions.
(a) 𝑓(𝑥) = 𝑥 2
(b) 𝑔(𝑥) = 𝑥 3
(c) 𝑕 𝑥 = 𝑥
(d) 𝑖 𝑥 = |𝑥|
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Math 1650 Lecture Notes
Jason Snyder, PhD.
§2.2
Graphs of Functions
Getting Information from the Graph of a Function
The graph of a function helps us picture the domain and range of the function on
the x-axis and y-axis as shown below.
Example 2 Finding the Domain and Range from a Graph
Below is the graph of the function 𝑓 𝑥 = 4 − 𝑥 2 . Find the domain and range
of 𝑓.
y
2
1
x
-2
-1
1
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2
Math 1650 Lecture Notes
Jason Snyder, PhD.
§2.2
Graphs of Functions
Graphing Piecewise Defined Functions
Example 3 Graph of a Piecewise Defined Function
Sketch the graph of the function
2
𝑓 𝑥 = 𝑥
2𝑥 + 1
𝑥≤1
𝑥>1
The greatest integer function is defined by
𝑥 = greatest integer which is less than 𝑥
Example 4 Graph of the Greatest Integer Function
Sketch the graph of 𝑓 𝑥 = 𝑥 .
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Math 1650 Lecture Notes
Jason Snyder, PhD.
§2.2
Graphs of Functions
The Vertical Line Test
The Vertical Line Test
A curve in the coordinate plane is the graph of a function if and only if no
vertical line intersects the curve more than once.
Example 5 Using the Vertical Line Test
Using the vertical line test, determine which of the following curves are graphs of
functions.
y
4
y
2
1.5
x
-3
1
-2
-1
1
2
3
-2
0.5
-4
x
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
y
y
8
5
6
x
-8
-6
-4
-2
2
4
6
8
4
2
-5
x
-8
-6
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-4
-2
2
4
6
8
Math 1650 Lecture Notes
Jason Snyder, PhD.
Example 6 Equations That Define Functions
Do the following equations define y as a function of x?
(a) 𝑦 − 𝑥 2 = 2
(b) 𝑥 2 + 𝑦 2 = 4
Homework
Due:_________________________
2 – 24 (even), 38 – 50 (even), 56, 62 – 72 (even)
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§2.2
Graphs of Functions