Functions and Their Graphs

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Functions and Their Graphs

• Section P.3

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After this lesson, you should be able to

• evaluate a function
• recognize the graphs of linear, squaring, cubing,
  square root, absolute value, rational, sine and
  cosine functions and graph them on a calculator
• find the domain and range of a function
• using the graphing calculator, graph piece-wise
  functions
• describe transformations of the basic functions
• find the zeroes of a function
• determine if a function is even or odd

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Definition of Function
Real-Valued Function of a Real Variable
Let X and Y be sets of real numbers.
A real-valued function f of a real variable x
from X to Y is a correspondence that assigns to
each number x in X
exactly one number y in Y.
The domain of f is the set X.
The number y
is the image of x under f
and is denoted by f(x).
The range of f is a subset of Y
and consists of all images
of numbers in X.
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Functions
F
range
domain
x
y
X
Y
Read about functions in your text on page 19 and
half of page 20.
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Example 1 from Text
Example 1 Evaluating a Function For the
function f defined by Evaluate each
expression.
a)
b)
c)
Refer to your text on page 20 to see these worked
out.
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Domain and Range of a function
More simply stated, we can say
Domain
x-values (or input values)
Range
y-values (or output values)
Try thinking of a function as a machine with a
crank. The domain values are those you put
into the machine. Turn the crank and the range
values are those that come out the machine. So
think about putting in domain values for x and
getting out the corresponding range values for y
(or f(x)). Youve done this a bunch of times
when you create a table of values.
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Domain of a Function
Example Define the domain of the function
ans domain is the set x x ? 2
Or in interval notation,
(On the next slide, youll find notes on interval
notation if you need a refresher.)
Example Define the domain of the function
ans domain is the set x 4 ? x ? 5
Or in interval notation,
Note In this case, the domain was restricted
from the beginning.
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Interval Notation-Refresher
Open interval (a, b)
Closed interval a, b
(a, b
a, b)
Note The interval is ALWAYS open where infinity
is part of the interval notation.
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Domain and Range of a Function
Example Graph the function on your calculator,
and then find the domain and range of the
function.
Ans
Domain
Range
In interval notation, domain of f is 1,
8) range of f is 0, 8)
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Domain and Range
Example Graph on your calculator and then
determine the domain and range of the function.
To graph this on your calculator, youll need to
graph both equations and restrict their domains.
Ill show you what the y menu should look like
in your calculator.
Note This is a piece-wise function.
You can find the inequality symbols using ? ? to
get the TEST menu
Graph of f(x)
Ans Domain all reals or (-8, 8)
Range 0, 8)
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Vertical Line Test
To determine if a graph is a function, a vertical
line test can be performed. In order for a graph
to be a function, a vertical line in any spot on
the graph can only intersect the graph at most
once.
Not a function
A function
A function
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Basic Functions
The graphs of the eight basic functions from page
22 in your text should be recognizable to you
from now on.
Absolute Value Function
Identity Function
Rational Function
Squaring Function
Cubing Function
Sine Function
Square Root Function
Cosine Function
Turn to page 22 in your text to see the graphs of
these functions OR graph them on your calculator.
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Some Graphing Refreshers-Absolute Value
To graph an absolute value function Start off
in the ? menu.
Then, select ? ? (which selects CATALOG), then
?.
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Some Graphing Refreshers-Trig
Keep in mind that when you want to graph a trig
function, youll want the mode to be in
radiansnot degrees!
To make sure your calculator is in radian
mode Select ?.
Make sure Radian is highlighted in black. If it
isnt, cursor on top of Radian, and hit ?.
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One-To-One Function
A function from X to Y is one-to-one if to each
y-value in the range, there corresponds exactly
one x-value in the domain.
Passes Horizontal Line Test
Note A function from X to Y is onto if its
range consists of all of Y.
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Polynomial Functions
The most common algebraic function is the
polynomial function.
n is a positive integer (the degree of the
polynomial function) ai are coefficients (an is
the leading coefficient and a0 is the constant
term)
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Rational Functions
The function f(x) is rational if
Polynomial, rational, and radical functions are
all algebraic functions. The other types are
trig and exponential functions.
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Elementary Functions
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Composite Functions
Let f and g be functions. The function given by
(f g)(x) f(g(x)) is called the composite of
f with g. The domain of f g is the set of all
x in the domain of g such that g(x) is in the
domain of f.
f g
g(x)
x
f(g(x))
f
g
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Example-Composite Functions
Example Given f(x) x2 1 and g(x) cos x,
find f g.
(Pythagorean Identity cos2x sin2x 1)
Example 4, on page 24 of the text, provides
another example.
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Tests for Even and Odd Functions
The function y f(x) is even if f(-x) f(x)
(has y-axis symmetry)
To test if a function is even, just apply the
test for y-axis symmetry, provided in section
P.1. If the graph has y-axis symmetry, then the
function is even.
The function y f(x) is odd if f(-x) -f(x)
(has origin symmetry)
To test if a function is odd, just apply the test
for origin symmetry, provided in section P.1. If
the graph has origin symmetry, then the function
is odd.
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Example 5 in Text
Example 5 Determine whether the function is
even, odd, or neither. Then find the zeros of
the function. a) b)
Refer to your text, Example 5, page 26 for the
corresponding work.
a) The function is odd since it is symmetric
about the origin.
b) The function is even since it is symmetric
about the y-axis.
The zeros of g occur when x (2n
1)?, where n is an integer. There are an
infinite number of zeroes to g since the function
is cyclical.
The zeros of f are x 0, -1, 1
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Homework
Section P.3 page 27 1, 3, 7, 17, 19, 23-27
odd, 29-41 odd, 55, 59, 61, 93 (a and b only)