# Lesson 2-4 Equations with Variables on Both Sides ```Lesson 1.2, pg. 138
Functions &amp; Graphs
Objectives: To identify relations and functions,
evaluate functions, find the domain and range of
functions, determine whether a graph is a function,
and graph a function.
Domain &amp; Range
• A relation is a set of ordered pairs.
• Domain: first components in the relation
(independent); x-values
• Range: second components in the relation
(dependent, the value depends on what the
domain value is); y-values
• Find the domain and range of the relation.
{(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )}
FUNCTIONS
• Functions are SPECIAL relations: A domain
element corresponds to exactly ONE range
element.
Every “x” has only one “y”.
Mapping – illustrates how each member of the domain
is paired with each member of the range (Note: List
domain and range values once each, in order.)
Draw a mapping for the following.
(5, 1), (7, 2), (4, -9), (0, 2)
x
y
0
4
5
7
-9
1
2
Is this
relation a
function?
See Example 2, page 150.
Determine whether each relation is a function:
A) {(1,2), (3,4), (5,6), (5,8)}
B) {(1,2), (3,4), (6,5), (8,5)}
Functions as Equations
Determine whether the equation defines y as a function of x.
a)
x2  y  4
b)
x2  y 2  4
1. Solve for y in terms of x.
2. If two or more values of y can be obtained for a given x,
the equation is not a function.
Determine if the equation defines y as a
function of x.
A) 2x + y = 6
B) x2 + y2 = 1
C) x2 + 2y = 10
Evaluating a Function
• Common notation: f(x) = function
• Evaluate the function at various values of x,
represented as: f(a), f(b), etc.
• Example: f(x) = 3x – 7
f(2) =
f(3 – x) =
If f(x) = x2 – 2x + 7, evaluate each of the
following.
• a) f(-5)
b) f(x + 4)
c) f(-x)
See Example 4, page 143 for additional practice.
Determine if a relation is a function
from the graph?
• Remember: to be a function, an x-value is
assigned ONLY one y-value .
• On a graph, if the x value is paired with MORE
than one y value there would be two points
directly on a vertical line.
• THUS, the vertical line test! If a vertical line drawn
on any part of your graph touches more than one
point, it is NOT the graph of a function.
Graphs of Functions
Step 1: Graph the relation. (Use graphing
calculator or pencil and paper.)
Step 2: Use the vertical line test to see if the
relation is a function.
• Vertical line test – If any vertical line passes
through more than one point of the graph,
the relation is not a function.
Determine if the graph is a function.
a)
y
b)
5
y
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x
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x
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-5
-5
5
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Here’s more practice.
c)
y
y
d)
x
x
Example
Analyze the graph.
y
f ( x)  x 2  3x  4
a. Is this a function?
b. Find f(4)
c. Find f(1)
d. For what value of x is f(x)=-4



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y

Find f(7).
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
(a) 0

(b) 1

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x
(c) 1
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(d) 2
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Can you identify domain &amp; range from the
graph?
• Look horizontally. What x-values are contained in
• Look vertically. What y-values are contained in the
• Write domain and range using interval or set-builder
notation.
• See Example 8, page 148.
Domain: set of all values of x
Range: set of all values of y
• Always write the domain and range in
domain and range from a graph.
• Use brackets [ or ] to show values that
are included in the graph.
• Use parentheses ( or ) to show values
that are NOT included in the graph.
Identify the function's domain and range from the graph
y
y
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Domain (-1,4]
Range [1,3)
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x
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Domain [3,)
Range [0,)


Example
y
Identify the Domain and Range from the graph.

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Example
y
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Identify the Domain and Range from the graph.
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x

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Example
y
Identify the Domain and Range from the graph.


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


x

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Find the Domain and Range.
y

D:(-, ) R:(-5,7]
(b) D:(-5,) R: (-, )
(c) D:(-, ) R: [-5,)
(d) D:[-, ] R: [-5,]

(a)
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x
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What is the difference in the two sets
below, and when should we use each
to describe the domain of a function?
{1,2,3,4}
[1,4]
Finding intercepts:
• x-intercept: where the function crosses the xaxis. What is true of every point on the x-axis?
The y-value is ALWAYS zero.
• y-intercept: where the function crosses the yaxis. What is true of every point on the yaxis? The x-value is ALWAYS zero.
• Can the x-intercept and the y-intercept ever
be the same point? YES, if the function
crosses through the origin!
We can identify x and y intercepts from a function's graph.
To find the x-intercepts, look for the points at which the graph
crosses the x axis. The y-intercepts are the points where the graph
crosses the y axis.
The zeros of a function, f, are the x values for which f(x)=0.
These are the x intercepts.
By definition of a function, for each value of x we can
have at most one value for y. What does this mean in terms
of intercepts? A function can have more than one x-intercept
but at most one y intercept.
y
Example

Find the x intercept(s). Find f(-4)
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x
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Example
y
Find the x and y intercepts. Find f(5).

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x
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Summary
• Domain = x values
• Range = y values
• Use the vertical line test to verify if a graph is
a function.
• To evaluate means to substitute and simplify.
• Intercepts – where function crosses the x-or yaxis
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