Functions

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Functions
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Click on a Topic

• What is a Function?
• Domains and Ranges
• Meet the Parents
• Transformations
• Composition of Functions

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The End (Now go practice what youve learned.
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Function Compositions
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So. . . how often do we really use a
composition of functions ?
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Well, have you ever used the output from one
operation on your calculator as the input of
another?
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I know I have.
Me Too.
Twice today!
Then you have used a composition of functions.
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Com-po-si-tion, n. Given two functions f and g,
the composite function,
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The Domain of the composition f(g(x)) is the set
of all x values in the domain of g such that g(x)
is in the domain of f.
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Example Find the domain of
The domain of g(x) all reals of f(x)
all reals except 0 of f(g(x)) all reals
except -2
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It is important to note that composition is not
a commutative operation. Order does matter.
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It is sometimes important to be able to see a
function as a composition of two others. (There
may be more than one possibility).
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Find another f(x) and g(x) such that
f(g(x)) h(x)
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Find another f(x) and g(x) such that
f(g(x)) h(x)
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Transformations
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Transformations allow us to take a parent
functions graph and move it up, down, left, or
right stretch or shrink it vertically or
horizontally, or flip it over an axis by means of
algebraic changes to the function.
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Translations of

• Slide UP K units
• Slide DOWN K units

Where k is a constant k gt 0.
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Parent
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To summarize, when a constant, k, is added or
subtracted to the parent after the fact
(outside of f(x) ) the result is a vertical
translation up or down of yf(x).
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Translations of

• Slide LEFT K units
• Slide RIGHT K units

Where k is a constant k gt 0.
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Parent
Notice that subtracting moves it to the right and
adding moves it to the left. This seems to be
illogical. Explanation follows.
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Remember The notation f (xk) or f (x-k)
indicates that k is added to or subtracted from
x before being evaluated in the function, f.
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So, to get the same output value as y1 , the x
substituted into y2 must be 3 less since we
immediately add 3 to it. For y3 it must be 3
more because we immediately subtract 3 from it.
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To summarize, when a constant, k, is added or
subtracted to the x before the fact (inside of
f(x) ) the result is a horizontal translation
left or right of yf(x) .
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Reflections of

• Over the x axis
• Over the y axis

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Parent
Notice how the Range is affected in the graph on
the left and how the Domain is affected in the
graph on the right.
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Reflections of

• Over the x axis
• Over the y axis

If a function is _______ then a reflection over
the x axis is the same graph as a reflection over
the y axis.
odd
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Reflections of

• Over the x axis
• Over the y axis

If a function is _______ then a reflection over
the y axis leaves the graph unchanged.
even
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Reflections of

• Over the x axis
• Over the y axis

If a reflection over the x axis leaves the graph
unchanged then____________________.
the graph is not a function
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Stretching/Shrinking

• Vertical Stretch
• Vertical Shrink

Where k is a constant k gt 1.
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Parent
Notice how the Range and amplitude (y data) are
directly affected in each graph, however, Domain,
and period (x data) are unaffected.
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Stretching/Shrinking

• Horizontal Stretch
• Horizontal Shrink

Where k is a constant k gt 1.
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Parent

• Horizontal Stretch
• Horizontal Shrink

Notice how the Period is affected in each graph
but Amplitude is not.
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Notice Dividing by k stretches the graph
horizontally and multiplying by k shrinks the
graph. Again, this may seem illogical, but. . .
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Remember The notation f (kx) or f (x/k)
indicates that x is multiplied by k or divided
by k before being evaluated in the function, f.
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So, to get the same output value as y1 the x
substituted into y2 must be 4 times as big since
we immediately divide it by 4 . For y3 it must
be ¼ as big because we immediately multiply by
4.
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What are The Parents?
These are the 20 Basic functions from which most
of the functions and graphs that you will be
working with this year will come. You need to
know these thoroughly.
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Line
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Absolute Value
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Quadratic
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Square Root
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Cubic
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Cube Root
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Hyperbola
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Exponential
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Logarithmic
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Semicircle
(Graph here appears to float above the axis,
but it should not.)
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Semi-hyperbola
(Graph here appears to float above the axis,
but it should not.)
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Sine Curve
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Cosecant Curve
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Tangent Curve
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Cosine Curve
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Secant Curve
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Cotangent Curve
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Arcsine Curve
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Arccosine
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Arctangent
(A true Arctangent graph was not available with
the graphing program that was used. The slope
at (0,0) should be 1, not undefined as it appears
here.)
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What is a Function?
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func-tion, n. A function f is a rule that
assigns to each element x in a set A exactly one
element, called f(x), in a set B.
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For us, A and B are sets of real numbers. Set A
is called the domain of the function.
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Notation f(x),read f of x,represents the
value of the function f at any x in the domain.
For example
f(2) represents the value of the function at
x2.
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The range of f is the set B of all possible
values of f(x) that can be obtained from the
function as x takes on all of the different
values in A, the domain.
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Notation When you see y f(x), it represents
a set of points, (x,y), whose y-coordinate for a
particular x is f(x).
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In y f(x), we call x the independent variable
and we call y the dependent variable.
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A Function can be given as
1. A Mapping or Arrow Diagram
-1 -2
1 2
A f B
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Note A Mapping would not represent a function
if one element from set A was mapped to two
different elements in set B.
-1 -2
1 2
A B
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A Function can be given
2. Numerically by A Table of Values
x -2 -1 0 1 2 3 4
f(x) -5 -3 -1 1 3 5 7
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Note A Table of Values would not represent y
as a function of x if an x value was assigned
more than one y value
x -2 1 0 1 2 3 4
y -5 -3 -1 1 3 5 7
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A Function can be given
3. Graphically
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Note A graph represents y as a function of x if
it passes the vertical line test. That is, any
vertical line drawn through the graph intersects
the graph in at most one point.
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Passing the vertical line test guarantees that
the definition of a function holds true for
each x, there is only one y.
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Note This graph does not represent y as a
function of x. It fails the vertical line test.
There exists at least one vertical line that
intersects the graph in more than one point.
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Failing the vertical line test means that the
definition of a function is not satisfied for
some x, there is more than one y.
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A Function can be given
4. Algebraically by an explicit formula. For
example
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A Function can be given
5. Piecewise by more than one formula
according to its different domain values.
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Why do we care whether a relation is a function
or not?
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Well, arent you glad that your calculator
functions the same way every time?
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Its operations are functions , so it always gives
the same output for a given
input.
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Odd and Even Functions
A function f is said to be even if f(x)f(-x) for
every x in the Domain of f. Visually, a graph
represents an even function if it has symmetry
with respect to the y-axis.
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The function pictured is an even function. It
has symmetry with respect to the y axis and
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The function pictured is an odd function. It has
symmetry with respect to (0,0) and
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Finding Domain and Range
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2 Ways to remember them

• Domain (set of x values) and Range (set of y
  values) are in alphabetical order, as are x and
  y.
• Domain is Input valuesand Range is Output
  values (also in alphabetical order).

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If a function is given as a Mapping Diagram,
then the Domain consists of all of the elements
of set A.
-1 -2
1 2
A f B
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If a function is given as a Mapping Diagram,
then the Range consists of all of the elements in
set B.
-1 -2
1 2
A f B
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If a function is given Numerically by A Table
of Values, the Domain is the set of x values.
x -2 -1 0 1 2 3 4
f(x) -5 -3 -1 1 3 5 7
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If a function is given Numerically by A Table
of Values, the Range is the set of f (x ) values.
x -2 -1 0 1 2 3 4
f(x) -5 -3 -1 1 3 5 7
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If a function is given Graphically, then the
Domain is the set of all x values on the
x-axis that
have part of the
graph on it, below it, or above it.
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The Domain for the graph pictured here is the set
of All Real Numbers.
Note All of the
x-axis has been
Used.
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If a function is given Graphically, then the
Range is the set of all y values on the
y-axis that
have part of the
graph On it, Left of it, or Right of it. (or
both)
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The Range for the graph pictured here is the set
of All Real Numbers.
Note All of the
y-axis has been
Used.
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Example This graph could be described
algebraically using a piecewise function. Notice
that the graph does
pass the vertical
line test.
(We will assume that the graph that goes off of
the page does so in a continuous manner.)
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Domain Range
Note These answers are given in Interval
Notation
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If a function is given Algebraically, to find
the domain, you must find the most complete set
of Real Numbers for which the function is
defined. (In other words, find all of the x
values that give back a real number when
substituted and evaluated in f(x). )
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The easiest thing to do is look for ways that
the function would fail to give a Real Number.
There are two problem
areas to
look for.
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The 2 Things to Look Out for

• Zeros in Denominators
• 2. Negatives under even roots

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You may see a combination of both of these things

For the numerator, -1 lt x lt 1.
For the denominator, x cant equal 0.5.
So, The Domain of this function must satisfy
both
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Interval Notation used for expressing a set of
Real numbers
Interval
Inequality a lt x lt b a lt x lt b a lt x
lt b
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Line Graph
Interval
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