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Functions
Simple to Advanced
Intro to Functions
Relation – mathematical rule for model of
numbers or data.
Function – relation in which each element
in the first set (x) has one and only one
element from the second set (y). (Passes
the vertical line test.)
Domain – set of inputs
Range – set of outputs
Pre-image – the set from which the domain
can be chosen.
Image - the set from which the range can be
chosen.
1 2
a b
4
3
d
f
5
c
e
g
h
One to one – one element of the domain
matches with one and only element in the range
and one element of the range matches with one
and only element in the domain. (Passes the
horizontal line test.)
Onto – range is all real numbers
Bounded – what happens at the extremes.
Continuous – can be drawn without lifting pencil
from paper.
Point of discontinuity – point where you lift
your pencil
Even function
1.
All exponents in polynomial equation are even. (inspection)
2.
Symmetric to the y-axis (graphically)
3.
f(-x) = f(x) (algebraically)
Odd function
1.
All exponents in polynomial equation are odd. (inspection)
2.
Symmetric to the origin (graphically)
3.
f(-x) = - f(x) (algebraically)
Family of Functions
yx
yc
yx
2
y  x3
Family of Functions
y
x
y x
y x
1
y
x
Family of Functions
y  loga x
y  ax
Representations of Functions
1.
2.
3.
4.
Verbally – sentence describing how input
variable is related to output variable.
Numerically – table or list of ordered pairs
matching input values with output values.
Graphically – points on a coordinate plane
where input is on horizontal axis and output is
on the vertical axis.
Algebraically – an equation in two variables.
Function?
Input value is the
number of math
classes taken in HS,
the output value is the
number of computer
classes taken in HS.
Input x
Output y
1
8
2
4
3 4
3 1
Piecewise Defined Function
 x 2  1,
f (x)  
 x  1,
x0
x0
Another Example
 x
 2  1,

f (x)   x  3,
3x  2,


x  2
2  x  1
x 1
Domains of Functions
Domain of a square root:
radicand ≥ 0
Domain of a rational equation:
denominator ≠ 0
Definition of Difference Quotient
((x+h), f(x+h))
m
f (x  h)  f (x)
xhx
(x, f(x))
f (x  h)  f (x)
m
xhx
f  x  h   f (x)
h
Difference Quotient
f (x  h)  f (x)
h
Find the difference quotient given:
f(x) = 2x2 2 – 3x + 4
2
2  x  h   3  x  h   4   2x  3x  4 
h
2x 2  4hx  2h 2  3x  3h  4  2x 2  3x  4
h
4hx  3h  2h 2
h
h(4x  3  2h)

h
 4x  3  2h, h  0
Application
Given a 10” x 12” piece of cardboard, find the dimensions of the box
with the maximum volume.
10 – 2x
x
12 -2x
v  lwh
x
v  12  2x 10  2x  x
Domain = (0, 5)
Therefore the box is: 8.4”x6.4”x1.8”
Increasing, Decreasing,
Constant
Relative Maxima, Minima
Relative Maxima
Relative Minima
Write Height or Length of rectangle as
a function of x or y.
y = - x2 + 4x - 1
h
L
x = ½ y2
H=-x2 +4x – 1 – 2
W = ½ y2
H = - x2 + 4x - 3
–0
Transformations

Rigid Transformations
 Shape
of graph stays the same!
 Vertical Shift h(x) = f(x) + c
Do as equation tells you
 f(x) + c move up c units
 f(x) - c move down c units

 Horizontal
Shift h(x) = f(x – c)
Do the opposite of what equation tells you
 f(x – c) move right c units
 f(x + c) move left c units

Transformations

Non-rigid Transformations
 Shape
of the graph changes
 Vertical Stretch (shrink) h(x) = cf(x)



Do as equation tells you multiply or divide by c
c > 1 vertical stretch multiply all y values by c
c < 1 vertical shrink multiply all y values by c
 Horizontal



Stretch (shrink) h(x) = f(cx)
Do opposite of what equation says
f(cx) divide all x values by c
f(x/c) multiply all x values by c
Remember!!


Operations on the y-values—Do what the
equation tells you!
Operations on the x-values—Do the
opposite of what the equation tells you!!
Reflections
Reflection across the x-axis:
h(x) = - f(x)
 Reflection across the y-axis:
h(x) = f(- x)

Transformations
f(x)
f(x) + 2
- f(x)
f(x + 3)
How do you sketch a graph
without a calculator?
Sketch f(x) = (x – 2)(x + 3)2(x + 1)3
What do we need in order to graph?
1. x-intercepts
2. y-intercept
3. End behavior of graph
4. Behavior of x-intercepts
f(x) = (x – 2)(x + 3)2(x + 1)3
What do we need in order to graph?
1. x-intercepts
(2, 0) (- 3, 0) (- 1, 0)
y-intercept
(0, - 18)
3. End behavior of graph
x6 starts high ends high
4. Behavior of x-intercepts
2.



x slashes
x2 bounces
x3 meanders (snake-like)
f(x) = (x – 2)(x + 3)2(x + 1)3
f(x) = (x – 2)(x + 3)2(x + 1)3