Transcript Lesson 1.3, page 154 More on Functions Objectives To find the difference quotient. Understand and use piecewise functions Identify intervals on which a function increases,

Lesson 1.3, page 154
More on Functions
Objectives
To find the difference quotient.
Understand and use piecewise functions
Identify intervals on which a function increases, decreases,
or is constant.
Use graphs to locate relative maxima or minima.
Identify even or odd functions & recognize the symmetries.
Graph step functions.
REVIEW of Lesson 1.2
Reminder: Domain Restrictions
For FRACTIONS:
 No zero in denominator!
7
ex.
 undefined
0
For EVEN ROOTS:
 No negative under even root!
ex. x  2 ,
4
x 2
Find the domain of each (algebraically)and
write in interval notation.
a) f ( x)  | 3 x  2 |
1
b) f ( x ) 
x4
c) g ( x)  x  7
d ) f ( x)  7  x
Functions & Difference Quotients
Useful in discussing the rate of change of
function over a period of time
 EXTREMELY important in calculus
(h represents the difference in two x values)
 DIFFERENCE QUOTIENT FORMULA:

f ( x  h)  f ( x )
h
Difference Quotient
The average rate of change (the slope of the secant line)
If f(x) = -2x2 + x + 5, find and
simplify each expression.

A) f(x+h)
If f(x) = -2x2 + x + 5, find and
simplify each expression.

B)
f ( x  h)  f ( x )
h
Your turn: Find the difference
quotient: f(x) = 2x2 – 2x + 1
f ( x  h)  2( x  h) 2  2( x  h)  1
f ( x  h)  2( x 2  2 xh  h 2  2 x  2h  1
f ( x  h)  2 x 2  4 xh  2h 2  2 x  2h  1
f ( x  h)  f ( x) 2 x 2  4 xh  2h 2  2 x  2h  1  (2 x 2  2 x  1)

h
h
PIECEWISE FUNCTIONS
Piecewise function – A function that is defined
differently for different parts of the domain; a
function composed of different “pieces”
Note: Each piece is like a separate function with
its own domain values.
 Examples: You are paid $10/hr for work up to
40 hrs/wk and then time and a half for
overtime.

10 x, x  40
f ( x)  

15 x, x  40
See Example 3, page 169.
Check Point 2:
20 if 0  t  60
Use the function C (t )  



20

0.40(
t

60)
if
t

60


to find and interpret each of the folllowing:
 a) C(40)
b) C(80)
Graphing Piecewise Functions



Draw the first graph on the coordinate
plane.
Be sure to note where the inequality
starts and stops. (the interval)
Erase any part of the graph that isn’t
within that interval.
Graph
See p.1015
3x
f ( x)  
x  3
y
10
9
8
for more
problems.
if x  1
if x  1
7
6
5
4
3
2
1
10  9  8 7 6  5 4 3 21
3 4 5 6 7 8 9 10
1 1 2
2
3
4
5
6
7
8
9
10
x
Describing the Function


A function is described by intervals,
using its domain, in terms of x-values.
Remember:
 refers to "positive infinity"
 refers to "negative infinity"
Increasing and Decreasing
Functions

Increasing: Graph goes “up” as you move
from left to right.
x  x , f (x )  f (x )
1
2
1
2

Decreasing: Graph goes “down” as you
move from left to right. x  x , f ( x )  f ( x )
1
2
1
2

Constant: Graph remains horizontal as
you move from left to right.
x1  x2 , f ( x1 )  f ( x2 )
Increasing and Decreasing
Constant
Increasing and Decreasing
See Example 1, page 166.

Check Point 1 – See middle of
page 166.
Find the Intervals on the Domain in which the
Function is Increasing, Decreasing, and/or
Constant
Relative Maxima and Minima



based on “y” values
maximum – “peak” or highest value
minimum – “valley” or lowest value
Relative Maxima and Relative
Minima
Even & Odd Functions & Symmetry


Even functions are those that are
mirrored through the y-axis. (If –x
replaces x, the y value remains the
same.) (i.e. 1st quadrant reflects into the
2nd quadrant)
Odd functions are those that are mirrored
through the origin. (If –x replaces x, the y
value becomes –y.) (i.e. 1st quadrant
reflects into the 3rd quadrant or over the
origin)
See Example 2, page 167.

Determine whether each function is even,
odd, or neither.
a) f(x) = x2 + 6
b) g(x) = 7x3 - x
Determine whether each function is
even, odd, or neither.
c) h(x) = x5 + 1
Your turn: Determine if the function
is even, odd, or neither.
f ( x)  2( x  4)  2x
2
a)
b)
c)
Even
Odd
Neither
2