Transcript Document

CHAPTER 1:
Graphs, Functions,
and Models
1.1
1.2
1.3
1.4
1.5
1.6
Introduction to Graphing
Functions and Graphs
Linear Functions, Slope, and Applications
Equations of Lines and Modeling
Linear Equations, Functions, Zeros and Applications
Solving Linear Inequalities
Copyright © 2009 Pearson Education, Inc.
1.2
Functions and Graphs
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Determine whether a correspondence or a
relation is a function.
Find function values, or outputs, using a
formula or a graph.
Graph functions.
Determine whether a graph is that of a
function.
Find the domain and the range of a function.
Solve applied problems using functions.
Copyright © 2009 Pearson Education, Inc.
Function
A function is a correspondence between a first set,
called the domain, and a second set, called the
range, such that each member of the domain
corresponds to exactly one member of the range.
It is important to note that not every correspondence
between two sets is a function.
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Example
Determine whether each of the following
correspondences is a function.
a.
–6
6
–3
3
0
36
9
0
This correspondence is a function
because each member of the
domain corresponds to exactly one
member of the range. The
definition allows more than one
member of the domain to
correspond to the same member of
the range.
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Example
Determine whether each of the following
correspondences is a function.
b.
Helen Mirren
The Queen
Jennifer Hudson
Blood Diamond
Dreamgirls
Leonardo DiCaprio
The Departed
Jamie Foxx
This correspondence is not a function because there
is one member of the domain (Leonardo DiCaprio)
that is paired with more than one member of the
range (Blood Diamond and The Departed).
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Relation
A relation is a correspondence between the first
set, called the domain, and a second set, called
the range, such that each member of the domain
corresponds to at least one member of the range.
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Example
Determine whether each of the following relations
is a function. Identify the domain and range.
a. {(9, –5), (9, 5), (2, 4)}
Not a function. Ordered pairs
(9, –5) and (9, 5) have the
same first coordinate and
different second coordinates.
Domain is the set of first coordinates: {9, 2}.
Range is the set of second coordinates: {–5, 5, 4}.
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Example (continued)
Determine whether each of the following relations
is a function. Identify the domain and range.
b. {(–2, 5), (5, 7), (0, 1), (4, –2)}
Is a function. No two
ordered pairs have the same
first coordinate and different
second coordinates.
Domain is the set of first coordinates: {–2, 5, 0, 4}.
Range is the set of second coordinates: {5, 7, 1, –2}.
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Example (continued)
Determine whether each of the following relations
is a function. Identify the domain and range.
b. {(–5, 3), (0, 3), (6, 3)}
Is a function. No two
ordered pairs have the same
first coordinate and different
second coordinates.
Domain is the set of first coordinates: {–5, 0, 6}.
Range is the set of second coordinates: {3}.
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Notation for Functions
The inputs (members of the domain) are values of x
substituted into the equation. The outputs (members of
the range) are the resulting values of y.
f (x) is read “f of x,” or “f at x,” or “the value of f at x.”
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Example
A function is given by f(x) = 2x2 – x + 3. Find each of
the following.
a. f (0) b. f (–7) c. f (5a) d. f (a – 4)
a. f (0)
f (0) = 2(0)2 – 0 + 3 = 0 – 0 + 3 = 3
b. f (–7)
f (–7) = 2(–7)2 – (–7) + 3 = 2 • 49 + 7 + 3 = 108
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Example (continued)
A function is given by f(x) = 2x2 – x + 3. Find each of
the following.
a. f (0) b. f (–7) c. f (5a) d. f (a – 4)
c. f (5a)
f (5a) = 2(5a)2 – 5a + 3 = 2 • 25a2 – 5a + 3
= 50a2 – 5a + 3
d. f (a – 4)
f (a – 4) = 2(a – 4)2 – (a – 4) + 3
= 2(a2 – 8a + 16) – a + 4 + 3
= 2a2 – 16a + 32 – a + 4 + 3
= 2a2 – 17a + 39
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Graphs of Functions
We graph functions the same way we graph equations.
We find ordered pairs (x, y), or (x, f (x)), plot the
points and complete the graph.
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Example
Graph f (x) = x2 – 5 .
Make a table of values.
x
f (x) (x, f (x))
–3
4
(–3, 4)
–2
–1 (–2, –1)
–1
0
1
–4
–5
–4
(–1, –4)
(0, –5)
(1, –4)
2
3
–1
4
(2, –1)
(3, 4)
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Example (continued)
Graph f (x) = x3 – x .
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Example (continued)
Graph f x   x  4 .
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Example
For the function f (x) = x2 – 5, use the graph to find
each of the following function values.
a. f (3)
b. f (–2)
a. Locate the input 3 on
the horizontal axis,
move vertically (up)
to the graph of the
function, then move
horizontally (left) to
find the output on the
vertical axis.
f (3) = 4
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Example
For the function f (x) = x2 – 5, use the graph to find
each of the following function values.
a. f (3)
b. f (–2)
b. Locate the input –2 on
the horizontal axis,
move vertically (down)
to the graph, then move
horizontally (right) to
find the output on the
vertical axis.
f (–2) = –1
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Vertical-Line Test
If it is possible for a vertical line to cross a
graph more than once, then the graph is not
the graph of a function.
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Example
Which of graphs (a) - (c) (in red) are graphs of
functions?
Yes.
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No.
No.
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Example (continued)
Which of graphs (d) - (f) (in red) are graphs of
functions? In graph (f), the solid dot shows that (–1, 1)
belongs to the graph. The open circle shows that (–1, –2)
does not belong to the graph.
No.
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Yes.
Yes.
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Finding Domains of Functions
When a function f whose inputs and outputs are real
numbers is given by a formula, the domain is
understood to be the set of all inputs for which the
expression is defined as a real number. When an input
results in an expression that is not defined as a real
number, we say that the function value does not exist
and that the number being substituted is not in the
domain of the function.
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Example
Find the indicated function values and determine whether
the given values are in the domain of the function.
a. f (1)
1
f (x) 
x3
1
1
f (1) 

1 3
2
Since f (1) is defined, 1 is in the domain of f.
b. f (3)
1
1
f (3) 

3 3 0
Since division by 0 is not defined, f (3) does not exist
and, 3 is in not in the domain of f.
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Example
3x 2  x  7
.
Find the domain of the function h(x)  2
x  2x  3
Solution:
We can substitute any real number in the numerator,
but we must avoid inputs that make the denominator
0. Solve x2 + 2x – 3 = 0.
(x + 3)(x – 1) = 0
x + 3 = 0 or x – 1 = 0
x = –3 or
x=1
The domain consists of the set of all real numbers
except –3 and 1, or {x|x ≠ –3 and x ≠ 1}.
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Visualizing Domain and Range
Keep the following in mind regarding the graph of
a function:
Domain = the set of a function’s inputs, found on
the horizontal axis;
Range = the set of a function’s outputs, found on
the vertical axis.
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Example
Graph the function. Then estimate the domain and range.
f ( x)  x  4
Domain = [–4, ∞)
Range = [0, ∞)
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