Lesson 1.3, page 154More on Functions

Objectives To find the difference quotient.Understand and use piecewise functionsIdentify 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 1.2Reminder: Domain Restrictions

For FRACTIONS: No zero in denominator!

For EVEN ROOTS: No negative under even root!

ex undefined7

.0

ex x x4. 2 , 2

Find the domain of each (algebraically)and write in interval notation.

) ( ) | 3 2 |

1) ( )

4

) ( ) 7

) ( ) 7

a f x x

b f xx

c g x x

d f x 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:

h

xfhxf )()(

Difference QuotientThe 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)

h

xfhxf )()(

Your turn: Find the difference quotient: f(x) = 2x2 – 2x + 1

2

2 2

2 2

2 2 2

( ) 2( ) 2( ) 1

( ) 2( 2 2 2 1

( ) 2 4 2 2 2 1

( ) ( ) 2 4 2 2 2 1 (2 2 1)

f x h x h x h

f x h x xh h x h

f x h x xh h x h

f x h f x x xh h x h x x

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 , 40( )

15 , 40

x xf x

x x

See Example 3, page 169. Check Point 2:

Use the function

to find and interpret each of the folllowing: a) C(40) b) C(80)

20 if 0 60( )

20 0.40( 60) if 60

tC t

t t

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

1 2 3 4 5 6 7 8 9 1010 9 8 7 6 5 4 3 2 1 123456789

10

123456789

10y

x

3 1( )

3 1

x if xf x

x if x

See p.1015

for more problems.

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.

Decreasing: Graph goes “down” as you move from left to right.

Constant: Graph remains horizontal as you move from left to right.

)()(, 2121 xfxfxx

)()(, 2121 xfxfxx

)()(, 2121 xfxfxx

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.

a) Even

b) Odd

c) Neither

22 2)4(2)( xxxf