REVIEW Reminder: Domain Restrictions For FRACTIONS: n No zero in denominator! For EVEN ROOTS: n No...
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REVIEWReminder: Domain Restrictions
For FRACTIONS: No zero in denominator!
For EVEN ROOTS: No negative under radical!
ex undefined7
.0
ex x x4. 2 , 2
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Review: Find the domain of each and write in interval notation.
) ( ) | 3 2 |
1) ( )
4
a f x x
b f xx
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Review: Find the domain of each and write in interval notation.
) ( ) 7
) ( ) 7
c g x x
d f x x
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More 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.
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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
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Difference QuotientThe average rate of change (the slope of the secant line)
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If f(x) = -2x2 + x + 5, find and simplify
A) f(x + h)
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If f(x) = -2x2 + x + 5, find and simplify
B)
h
xfhxf )()(
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c) Your turn: Find the difference quotient: f(x) = 2x2 – 2x + 1
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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( )
10(40) 15( 40), 40
x xf x
x x
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Example – Cell Phone Plan($20 for < 1 hour plus 40 cents per minute over 60)
Use the function
to find and interpret each of the following: d) C(40) e) C(80)
20 if 0 60( )
20 0.40( 60) if 60
tC t
t t
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Graphing Piecewise Functions
Graph each “piece” on the same coordinate plane.
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Functions Defined Piecewise Graph the function defined as:
.
2
3 for 0
( ) 3 for 0 2
1 for 22
x
f x x x
xx
y
x
10
10
-10
-10
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f(x) = 3, for x 0
f(x)= 3+ x2, for 0< x 2
( ) 1 for 22
xf x x
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Piecewise Graphs Extra Example4 for 2
( ) 1 for -2 3
for x 3
x
f x x x
x
y
x
10
10
-10
-10
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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"
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Increasing and Decreasing Functions
Describe by observing the x-values. 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
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Increasing and Decreasing
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Constant
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Increasing and Decreasing
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Find the Intervals on the Domain in which the Function is Increasing, Decreasing, and/or Constant
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Relative Maxima and Minima
based on “y” values
maximum – “peak” or highest value minimum – “valley” or lowest value
We say, “It has a relative maximum at (x-value) and the maximum is (y- value).”
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Relative Maxima and Relative Minima
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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)
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Example Determine whether each function is even,
odd, or neither.f) f(x) = x2 + 6 g) g(x) = 7x3 - x
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Determine whether each function is even, odd, or neither.
h) h(x) = x5 + 1
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i) Your turn: Determine if the function is even, odd, or neither.
a) Even
b) Odd
c) Neither
22 2)4(2)( xxxf