1.2 Functions and Their Properties
A _________ from a set D to a set R is a rule that assigns to
every element in D a unique element in R.
*Note: ______ cannot repeat.
The set D of all input values (x-values) is the ________ of the
function.
The set R of all output values (y values) is the _________ of the
function.
Examples
Determine if the following are functions.
1.) A = {(0,1) , (-9,10) , (3,10)}
2.) A = {(0,1) , (0,10) , (3,10)}
Jul 31­3:30 PM
Note: f(x) is fancy for ____.
Determine whether the formula determines y as a function of x.
If not, explain.
1.) y = x2
2.) x = y2
3.) y = x2 ± 4
4.) y = 2x - 1
Is there a way to look at a graph and determine if it's a function?
Vertical Line Test: A graph [set of points (x,y)] in the xy-plane
defines y as a function of x if and only if no vertical line
intersects the graph in more than one point.
Jul 31­3:38 PM
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Examples Use the vertical line test to determine whether the
curve is a graph of a function.
1.)
2.)
3.)
4.)
Jul 31­3:44 PM
Domain
-The domain of a polynomial is always
____ or ____.
Examples of polynomials:
-Restrictions to the domain come when
we have _______ ________ or
_______ _____ ________.
Examples:
Examples: Find the domain of the function algebraically and
support your answer graphically.
1.) f(x) = x3 + 1
2.) g(x) = 5
x-2
Jul 31­3:51 PM
2
x+1
3.) h(x) =
(x + 1)(x - 1)
4.) m(x) = √x
5.) p(x) = √x - 2
6.) j(x) =
√5 - x
(x + 2)(x2 + 3)
Jul 31­3:59 PM
Range
Examples: Find the range of the
functions.
1.) f(x) = x2 + 4
2.) g(x) = 5 - x
3.) h(x) = √x + 2 + 3
4.) m(x) =
x2
4 - x2
Jul 31­4:04 PM
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at a point if the graph does not
A function is
break at that point.
Note: You can draw a continuous function without lifting your
writing utensil.
Removable discontinuity: (Holes) The discontinuity can be
patched by redefining the output value so as to plug the hole.
Nonremovable discontinuity: (Jump discontinuity and
asymptotes) The discontinuity is impossible to patch by
redefining the output value.
Jul 31­4:18 PM
Jul 31­4:28 PM
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Examples Graph the functions determine any points of
discontinuity. If there are any discontinuities, tell whether they
are removable or nonremovable. Note: It might be helpful to look
at the algebraic representations, as well.
1.) f(x) = 5
x
2.) g(x) = x2 + x
x
3.) g(x) = -1, x > 0
1, x ≤0
4.) k(x) = 3
x-1
Jul 31­4:25 PM
Increasing, Decreasing and Constant Functions
Examples: Determine the intervals where the functions are
increasing, decreasing or constant.
1.)
2.) g(x) = x + 1 + x - 1 - 3
Jul 31­4:51 PM
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Boundedness
Jul 31­4:57 PM
Examples Determine whether the function is bounded above,
bounded below, or bounded on its domain.
1.) y = 19
2.) y = x2 + 5
3.) y = (1/2)x
4.) y = x5 + x - 1
5.) y = -√4-x2
Jul 31­8:41 PM
6
Extrema
of a function f is a value f(c) that is
A _____
greater than or equal to all range values of f on some open
interval containing c. If f(c) is greater than or equal to all range
values of f, then f(c) is the ________(or _______ ________)
value.
A _____ _________ of a function f is a value f(c) that is less
than or equal to all range values of f on some open interval
containing c. If f(c) is less than or equal to all range values of f,
then f(c) is the _______ or (_______ _________) value of f.
________.
Local extrema are also called
Jul 31­8:45 PM
Examples Identify the extrema.
1.)
2.)
3.) g(x) = x3 - 4x + 1
Jul 31­9:06 PM
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Symmetry
Symmetry with respect to (w.r.t) the y-axis
Example: f(x) = x2
Numerically
f(x)
x
-3
-2
-1
0
1
2
3
Graphically
Algebraically
For all x in the
domain of f,
f(-x) = f(x).
Functions with this
property are called
______ functions.
Aug 1­11:26 AM
Symmetry w.r.t the x-axis
Example: x = y2
Numerically
x
y
9
4
1
0
1
4
9
Graphically
Algebraically
Graphs with this
kind of symmetry
are not functions
(except f(x)=0),
but we can say
that _____ is on
the graph
whenever ____ is
on the graph.
Aug 1­11:35 AM
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Symmetry w.r.t the origin
Example: f(x) = x3
Numerically
x
f(x)
-3
-2
-1
0
1
2
3
Algebraically
For all x in the
domain of f,
f(-x) = -f(x).
Functions with
this property are
called _____
functions.
Graphically
Aug 1­11:41 AM
Examples State whether the function is odd, even or neither.
Support graphically and confirm algebraically.
2.) f(x) =
1.) h(x) = 1/x
3
1 + x2
3.) f(x) = x3 + 0.04x2 + 3
Aug 1­11:52 AM
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Asymptotes
-Vertical asymptotes: A vertical line (_____) that the function
approaches, but never reaches.
To find them, simplify the function completely.
Set the denominator = 0 and solve.
-Horizontal asymptotes: A horizontal line (_____) that the
function approaches, but never reaches.
Cases:
1.) If the top power is bigger, there are no H.A.
2.) If the top and bottom powers are the same, the H.A. is the
ratio of the leading coefficients.
3.) If the bottom power is bigger, the H.A. is y = 0.
Aug 1­11:54 AM
Examples Find all of the horizontal and vertical asymptotes.
1.) f(x) =
2.) g(x) = 5x + 2
x3 + 27
6x
2x + 1
4.) k(x) = 6x
3.) h(x) = 4x - 20
x2 - 25
5.) j(x) = (1.2)x - 3
Aug 1­12:01 PM
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End Behavior
Limit Notation: lim f(x) = ___ means as x goes to infinity, where
does f(x) go?
x--> ∞
lim f(x) = ___ means as x goes to negative
infinity, where does f(x) go?
x--> -∞
Examples: Determine the end behavior.
1.)
2.)
3.) f(x) =
4
1 + e-x
Aug 1­1:46 PM
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