Transcript Functions

Functions
Lesson 1.3
The Magic Box
• Consider a box that
receives numbers in
the top
• And alters them
somehow and sends a
(usually) different
number out the spout
14
3
-33
1
Today we look at the
For each of the number
mathematical
way of
combinations, can you figure
talking
about
the
out the "rule" which alters the
"rule"
number?
What is a Function?
• Definition:
A function is a rule
– Given X = { x1, x2, …} and Y = {y1, y2, …}
– Assign to each element of X a unique element
of Y
• The set X is the domain
– the set of all possible x values
• The set Y is the range
– the set of all resulting y values
Notation
• y = f(x) can be thought of as ..
– “y is the image of the function f at x”
or
– “y is the value of the function f at the point x”
• It is often read “y equals f of x”
– your instructor will often use the phrase
“f at x …”
– this reflects the above
Alternate Definition
• A function is a set of ordered pairs:
f(x) = { (x1, y1), (x2, y2), … }
• Where no two ordered pairs have the same first
element
• Which of these is a function?
 { (3,4), (6,0), (9,4), (7,2)}
 {(1, 2), (3, 4), (5, 6), (1, 7)}
 {(8,-1), (9, -1), (10, -1), (105, -1)}
Function Notation
• Normal notation:
f ( x)  2 x  7 x
2
• Functions can be defined/declared in your
calculator:
• Note the use of
functional
notation f(3) to
evaluate functions.
The -> is the STO> key
Piecewise Functions
• Some functions may be defined differently
for different portions of the domain.
 x  1 if x  5
f ( x)  
 x if x  5
• Your calculator can
also define piecewise
functions
Piecewise Functions
• Note the results when
the function is graphed:
– Actually there should be a gap between 6 and the
square root of 5
– Which of the above values should apply for the
function
– Try the F3, trace on the graph
Domain and Range
• Either the domain or range or both can be
restricted
– due to the nature of the function
• Consider f ( x)  x  1 g ( x)  x 2  5 x  6
x
• Determine the domain and range for each:
Domain and Range
• f(x)
1
f ( x)  x 
x
g ( x)  x 2  5 x  6
– Domain: all real numbers
• BUT … not zero … why?
– Range: y < -2 or y > 2
• g(x)
– Domain: x < -3 or x > -2 (why?)
– Range: y >= 0
Composition of Functions
• Basic Concept:
– value fed to first
function
– result fed to second
function
– end result taken from
second function
• Oft used notation: y = g(f(x))
or
g ( f ( x))  g f ( x)
Composition of Functions
• Example – given : f ( x)  2  x
1
g ( x)  2
x
• Try these
f(4) = ?
g(f(4)) = ?
f(g(-2)) = ?
Try defining the
functions on your
calculator and using
the notation !!
Assignment
• Lesson 1.3A
• Page 31
• Exercises: 3, 7, 11, 15, 21,
25, 27, 37, 43, 49
Defining a Graph
• The graph of a function is:
– the set of all points (x,y) which …
– satisfy the function y = f(x)
y  x3  6 x 2  x  30
Some Graphs are NOT Functions
• Which are not functions?
Use the “vertical line” test.
Intercepts
• We are often concerned with where the
graph intersects the axes
– x-intercepts => when f(x) = y = 0
– y-intercepts
=> when x = 0, f(0)
Reference Table of Functions
Page 27, Text
• Linear
• Quadratics
• Cubic
• Absolute
• Root functions
• Reciprocals
• Trig functions
Symmetry of Graphs
• Symmetric with the y-axis
– f(-x) = f(x) for all x
Called even
functions
Symmetry of Graphs
• Symmetric about the origin
– f(-x) = -f(x)
Called odd
functions
Note: There are many
functions which are
neither odd nor even
Transformation
• Shift a function up or down
y – k = f(x)
– k > 0 => up
– k < 0 => down
• Shift function right or left
y = f(x – h)
– h > 0 => right
– h < 0 => left
Transformation
• Define the function on the home screen
• Enter different versions of
f(x + h)
Try for the following
f(x)+k
function:
a*f(x)
3
f(-x)
.1x  .9 x  f ( x)
-f(x)
Assignment
• Lesson 1.3B
• Pg 31
• Exercises 51, 53, 57, 59, 63, 67, 69