Functions…
Click on a Topic:
• What is a Function?
• Domains and Ranges
• Meet the Parents
• Transformations
• Composition of Functions
The End
(Now go practice
what you’ve learned.
Function
Compositions
So. . . how
often do we
really use a
composition
of functions ?
Well, have you ever
used the output
from one operation
on your calculator
as the input of
another?
I know I
have.
Me Too.
Twice
today!
Then you have used a composition of
functions.
Com-po-si-tion, n.
Given two functions f
and g, the composite
function,
f g ( x)  f ( g ( x))
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.
Example: Find the domain of
f
g ( x)
1
Let f ( x)  and g ( x)  x  2
x
The domain of g(x): all reals
of f(x): all reals except 0
of f(g(x)): all reals except -2
It is important to note that
composition is not a commutative
operation. Order does matter.
If f ( x)  x  1 and g ( x)  x
f g ( x)  x  1 whereas
3
g f ( x) 

x 1

3
3
It is sometimes important to be
able to “see” a function as a
composition of two others.
(There may be more than one
possibility).
1
h( x )  2
x 1
1
f  x 
x 1
g ( x)  x
2
Find another f(x) and g(x) such that
f(g(x)) = h(x)
1
f  x 
x
1
h( x )  2
x 1
g ( x)  x  1
2
Find another f(x) and g(x) such that
f(g(x)) = h(x)
1
h( x )  2
x 1
1
f  x 
x 1
g ( x)  x  2
2
Return to Topics
Transformations
Transformations allow us to
take a “parent” function’s
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.
Translations of y  f ( x) :
• Slide UP K units
• Slide DOWN K
units
y  f ( x)  k
y  f ( x)  k
Where k is a constant k > 0.
Parent: y  x
y  x  3 (up)
y x
y  x  3 (down)
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
y=f(x).
Translations of y  f ( x) :
• Slide LEFT K units
• Slide RIGHT K
units
y  f (x  k)
y  f (x  k)
Where k is a constant k > 0.
Parent: y  x
y  x5
y  x 5
(left )
(right )
y x
Notice that subtracting moves it to the right and adding moves
it to the left. This seems to be illogical. Explanation follows.
Remember: The notation
f (x+k) or f (x-k) indicates
that k is added to or
subtracted from x before
being evaluated in the
function, f.
y1  x
y2  x  3
y3  x  3
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.
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
y=f(x) .
Reflections of y  f ( x) :
• Over the x axis
y   f ( x)
• Over the y axis
y  f ( x)
Parent:
y
y
x
x
y 
x
Notice how the Range is affected in the graph on the left and how
the Domain is affected in the graph on the right.
Reflections of y  f ( x) :
• Over the x axis
• Over the y axis
y   f ( x)
y  f ( x)
If a function is _______
odd then a
reflection over the x axis is the
same graph as a reflection over
the y axis.
Reflections of y  f ( x) :
• Over the x axis
• Over the y axis
y   f ( x)
y  f ( x)
If a function is _______
even then a
reflection over the y axis leaves
the graph unchanged.
Reflections of y  f ( x) :
• Over the x axis
• Over the y axis
y   f ( x)
y  f ( x)
If a reflection over the x axis
leaves the graph unchanged
the graph is not a function
then____________________.
Stretching/Shrinking:
y  f ( x)
• Vertical Stretch y  kf ( x )
• Vertical Shrink
1
y  f ( x)
k
Where k is a constant k > 1.
y  4sin x
( stretch)
Parent: y  sin x
1
y  sin x
4
Notice how the Range and amplitude (y data)
( shrink ) are directly affected in each graph, however,
Domain, and period (x data) are unaffected.
Stretching/Shrinking:
y  f ( x)
1
• Horizontal Stretch y  f ( x)
k
• Horizontal Shrink
y  f (kx)
Where k is a constant k > 1.
Parent: y  sin x
• Horizontal Stretch
x
y  sin
4
• Horizontal Shrink
y  sin 4 x
Notice how the Period is affected in each graph but Amplitude is not.
Notice: Dividing by k
stretches the graph
horizontally and
multiplying by k shrinks
the graph. Again, this may
seem illogical, but. . .
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.
y1  sin x
x
y2  sin
4
y3  sin 4 x
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.
Return to Topics
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.
Line:
f ( x)  x
Intercept : (0, 0)
slope  1
Domain : (, )
Range : (, )
Odd Function( Symmetry about origin )
Absolute Value:
f ( x)  x or
 x for x  o
f ( x)  
 x for x  0
Intercept and Minimum : (0, 0)
Domain : (, )
Range :  0,  
Even Function ( Symmetry about y  axis )
Quadratic:
f ( x)  x
2
Vertex and
Minimum : (0,0)
Domain : (, )
Range :  0,  
Even Function ( Symmetry about y  axis )
Square Root
f ( x)  x
Intercept : (0,0)
Domain :  0,  
Range :  0,  
Cubic
f ( x)  x
Intercept : (0,0)
Domain : (, )
Range : (, )
Odd Function (Symmetry about origin)
3
Cube Root
f ( x)  x
3
Intercept : (0,0)
Domain : (, )
Range : (, )
Odd Function ( Symmetry about origin)
Hyperbola:
1
f ( x) 
x
Domain : (,0)  (0, )
Range : (,0)  (0, )
Asymptotes : y  0 and x  0
Odd Function ( Symmetric about origin)
Symmetric about y  x
( Is its own inverse).
Exponential:
f ( x)  e
Intercept : (0,1)
Domain : (, )
Range : (0, )
Asymptote : y  0
x
Logarithmic:
f ( x)  ln x
Intercept : (1,0)
Domain :  0,  
Range :(, )
Inverse of f ( x)  e
x
Semicircle
f ( x)  1  x
x Intercepts : (1,0) and (1,0)
2
y Intercept : (0,1)
Domain :  1,1
Range :  0,1
(Graph here “appears” to float
above the axis, but it should not.)
Even Function (Symmetric to y  axis )
Semi-hyperbola:
f ( x)  x  1
2
Intercepts :  1,0  and (1,0)
Asymptotes : y  x and y   x
Domain :  , 1  1,  
Range :  0,  
(Graph here “appears” to float
above the axis, but it should not.)
Even Function ( Symmetry to y  axis )
Sine Curve
f ( x)  sin( x)
Period  2
Amplitude : 1
x Intercepts :( n ,0) for n  0,1, 2,...
y Intercept : (0,0)
Domain : (, )
Range :  1,1
Odd Function
Cosecant Curve
f ( x)  csc( x)
Period  2
Asymptotes : x   n
Domain : All reals except x  n for n  0,1,2,...
Range : (  1]  [1, )
Odd Function
Tangent Curve
f ( x)  tan( x)
Period  

Asymptotes : x   n
2
x Intercepts : ( n ,0)

Domain : All reals except x   n for n  0,1, 2,...
2
Range : (, )
Odd Function
Cosine Curve
f ( x)  cos( x)
Period  2
Amplitude : 1

x Intercepts :(  n ,0) for n  0,1, 2,...
2
y Intercept : (0,1)
Domain : (, )
Range :  1,1
Even Function
Secant Curve
f ( x)  sec( x)
Period  2

Asymptotes : x   n
2
y Intercept : (0,1)

Domain : All reals except x   n for n  0,1, 2,...
2
Range : (  1]  [1, )
Even Function
Cotangent Curve
f ( x)  cot( x)
Period  
Asymptotes : x   n

x Intercepts : (  n ,0)
2
Domain : All reals except x   n for n  0,1, 2,...
Range : (, )
Odd Function
Arcsine Curve
1
f ( x )  Sin ( x )
Intercept : (0, 0)
Domain :[ 1,1]
  
Range :   , 
 2 2
Odd Function
Arccosine:
1
f ( x)  Cos ( x)
x Intercept : (1, 0)

y Intercept : (0, )
2
Domain :[1,1]
Range :[0,  ]
Arctangent:
1
f ( x)  Tan ( x)
Intercept : (0, 0)
Asymptotes : y  

2
Domain : (, )
  
Range :   , 
 2 2
Odd Function
(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.)
Return to Topics
What is a
Function?
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.
For us, A and B are sets
of real numbers. Set A is
called the domain of the
function.
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 x=2.
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.
Notation: When you see
y = f(x), it represents a set
of points, (x,y), whose ycoordinate for a particular
x is f(x).
In y = f(x), we call x the
independent variable and
we call y the dependent
variable.
A Function can be given as:
1. A Mapping or Arrow Diagram:
1
-1
2
-2
A
f
B
Note: A Mapping would not
represent a function if one element
from set A was mapped to two
different elements in set B.
1
-1
2
-2
A
B
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
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
A Function can be given :
3. Graphically:
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.
Passing the vertical
line test guarantees
that the definition of a
function holds true…
for each x, there is
only one y.
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.
Failing the vertical
line test means that
the definition of a
function is not
satisfied…
for some x, there is
more than one y.
A Function can be given :
4. Algebraically– by an explicit
formula. For example:
f ( x) 
1
2x 1
A Function can be given :
5. “Piecewise”– by more than one
formula according to its different
domain values.
 x for x  0
f ( x)  
 x for x  0
2
Why do we
care whether
a relation is a
function or
not?
Well, aren’t you glad
that your calculator
“functions” the
same way
every time?
Its operations are
functions , so it
always gives the
same output
for a given
input.
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.
The function
pictured is an
even function.
It has symmetry
with respect to
the y axis and:
f ( x)  x  4 x
4
f ( x)  ( x)  4( x)
4
2
 x  4 x  f ( x)
4
2
2
The function
pictured is an
odd function. It
has symmetry
3
with respect to
f ( x)  x  4 x
(0,0) and: f ( x)  ( x)3  4( x)
 x  4x
3
Return to Topics
 ( x  4 x )   f ( x )
3
Finding Domain
and Range
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).
If a function is given as a Mapping
Diagram, then the Domain consists
of all of the elements of set A.
1, 2
1
-1
2
-2
A
f
B
If a function is given as a Mapping
Diagram, then the Range consists
of all of the elements in set B.
1, 2
1
-1
2
-2
A
f
B
If a function is given Numerically
by A Table of Values, the Domain is
the set of x values.
2, 1,0,1, 2,3, 4
x
-2
-1
0
1
2
3
4
f(x) -5
-3
-1 1
3
5
7
If a function is given Numerically
by A Table of Values, the Range is
the set of f (x ) values.
5, 3, 1,1,3,5,7
x
-2
-1
0
1
2
3
4
f(x) -5
-3
-1 1
3
5
7
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.
The Domain for the graph pictured
here is the set of All Real Numbers.
Note: All of the
x-axis has been
“Used”.
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)
The Range for the graph pictured
here is the set of All Real Numbers.
Note: All of the
y-axis has been
“Used”.
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.)
Domain:
Range:
 , 1  1, 
 , 6 
Note: These
answers are
given in
“Interval
Notation”
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). )
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.
The 2 Things to Look Out for:
1. Zeros in
Denominators
2. Negatives
under even
roots
1
f ( x) 
2x  3
x can ' t be  1.5
f ( x)  5  2 x
x must be less than
or equal to 2.5
4
You may see a combination of
both of these things :
1 x
f ( x) 
2x 1
2
For the numerator,
-1 < x < 1.
For the
So, The Domain of this
denominator, x function must satisfy
can’t equal 0.5. both:
Return to Topics
 1 1 

1,

,1


 2   2 
Interval Notation – used for
expressing a set of Real numbers:
Inequality:
Interval:
a<x<b
[ a, b]
a<x<b
( a, b)
[ a, b)
a<x<b
Line Graph:
a
a
b
a
Return
Interval:
[ a,  )
(, a)  {b}
(, a)  (a, )