Transcript 10.6 - DRS & Company

Chapter 10
Graphing
Equations and
Inequalities
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
10.6
Introduction to
Functions
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Vocabulary
• An equation in 2 variables defines a relation
between the two variables.
• A set of ordered pairs is also called a
relation.
• The domain is the set of x-coordinates of the
ordered pairs.
• The range is the set of y-coordinates of the
ordered pairs.
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Example
Find the domain and range of the relation {(4,9), (–4,9),
(2,3), (10, –5)}.
• Domain is the set of all x-values; {4, –4, 2, 10}.
• Range is the set of all y-values; {9, 3, –5}.
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Functions
Some relations are also functions.
A function is a set of order pairs in which each
x-coordinate has exactly one y-coordinate.
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Example
Is the relation{(4,9), (–4,9), (2,3), (10, –5)}, also a
function?
Since each element of the domain is paired with only
one element of the range, it is a function.
Note: It’s okay for a y-value to be assigned to more
than one x-value, but an x-value cannot be assigned to
more than one y-value (has to be assigned to ONLY
one y-value).
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Vertical Line Test
Graphs can be used to determine if a relation is a
function.
Vertical Line Test
If a vertical line can be drawn so that it
intersects a graph more than once, the
graph is not the graph of a function.
(If no such vertical line can be drawn, the
graph is that of a function.)
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Example
y
Use the vertical line
test to determine
whether the graph to
the right is the graph of
a function.
Since no vertical line
will intersect this
graph more than once,
it is the graph of a
function.
x
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Example
y
Use the vertical line
test to determine
whether the graph to
the right is the graph of
a function.
x
Since no vertical line
will intersect this
graph more than once,
it is the graph of a
function.
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Example
y
Use the vertical line test
to determine whether the
graph to the right is the
graph of a function.
Since vertical lines can
be drawn that intersect
the graph in two points,
it is NOT the graph of a
function.
x
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Vertical Line Test
Since the graph of a linear equation is a line,
all linear equations are functions, except those
whose graph is a vertical line.
Thus, all linear equations are functions
except those of the form x = c, which are
vertical lines.
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Using Function Notation
• The variable y is a function of the variable x. For each
value of x, there is only one value of y. Thus, we say the
variable x is the independent variable because any value
in the domain can be assigned to x. The variable y is the
dependent variable because its value depends on x.
• We often use letters such as f, g, and h to name functions.
For example, the symbol f(x) means function of x and is
read “f of x.” This notation is called function notation.
• We can use function notation to write the equation
y = –3x + 2 as f(x) = –3x + 2.
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Helpful Hint
Note that f(x) is a special symbol in
mathematics used to denote a function. The
symbol f(x) is read “f of x.” It does not mean
f • x (f times x).
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Function Notation
When we want to evaluate a function at a particular
value of x, we substitute the x-value into the notation.
For example, f(2) means to evaluate the function f when
x = 2. So we replace x with 2 in the equation.
For our previous example when f(x) = –3x + 2,
f(2) = –3(2) + 2 = –6 + 2 = –4.
When x = 2, then f(x) = –4, giving us the order pair
(2, –4).
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Example
Given g(x) = x2 – 2x, find g(–3). Then write down
the corresponding ordered pair.
g(–3) = (–3)2 – 2(–3) = 9 – (–6) = 15.
The ordered pair is (–3, 15).
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Example
y
Find the domain and
the range of the
function.
Domain: –3 ≤ x ≤ 4]
Domain
Range
x
Range: –4 ≤ x ≤ 2]
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Example
y
Find the domain and
the range of the
function graphed.
Range
x
Domain: all real
numbers
Domain
Range: y ≥ –2
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