functions are used to model mathematical situations. examples: a=πr 2 the area of a circle is a...
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•Functions are used to model mathematical situations.
Examples:A=πr2 The area of a circle is a
function of its radiusC=5/9(F – 32) °C as a function of
degrees °F
Functions: What the f?
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Function notation
y = 1 - x2 f(x) = 1 - x2
“y is a function of x” function notation
f(x) is read “f of x”
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Evaluating functions
f(x) = 1 - x2
What is f(-1)?In other words, evaluate the function
f(x) at x = -1.f(-1) = 1 – (-1)2 = 1 – 1 = 0
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Domain: For what values of x is the function defined?
Range: For what values of y is the function defined?
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Basic examples
f(x)=ax2+bx+c {all real numbers} g(x)=1/x {x: x 0} *denominator cannot be 0
h(x)=√x {x: x ≥ 0}
v(x)=ln(x) {x: x > 0}
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**** Notation time-out******
In mathematics,: means “such that”
Ex: {x : x 0} means “the set of all x’s such that x
0”
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Domain of
y = 2x + 3
{x: 2x + 3 0}
“all x’s such that 2x+3 is greater than or equal to 0”
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Domain of
y =
{x: 2x + 3 0}
Or {x: x -3/2}
1 2x +
3
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Domain of
y = 2x + 3All real numbers
3
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Domain of
y = ln(2x + 3)
{ x: 2x + 3 > 0 }
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Domain of
y =
{ x: 2x + 3 > 0 }
1 2x + 3
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Assignment Ap.27/ 1,2,(3-23)odd,37
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SAT Problem of the Day!
“700 on the SAT math!!! Heck yes, I am going to UCLA!”
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* A more CONCISE way to describe sets.
** Is used interchangeably with set notation to express domain and range.
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{x:…} INTERVAL
x 3
x 3
x 3
x 3
x 3
(- , 3)
(- , 3]
(3, )
[3, )
(- , 3) & (3, )
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-2 x 3
-2 x 3
-2 x 3
-2 x 3
All reals (- , )
(- 2 , 3][- 2 , 3)
(- 2 , 3)
[- 2 , 3]
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MIX AND MATCH!!!
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Assignment BIgnore the directions. Instead…Find the domain of each function
and write it using a) set notation and b) interval notation.
p.27-28/ 4,6,8,20,30,34
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SAT Problem of the Day!
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y = |x|
Absolutevalue
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y = x2
parabola
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y = x
Evenroot
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y = x3
Oddroot
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1
y = e x
Exponentialgrowth
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1
y = e - x
Exponentialdecay
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1
y = ln x
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y = 1/x
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y = 1/x2
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y = x2 - 2
y = x2
y = x2 + 3
+ move up - move
down
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y = |x|
y = |x - 6| y = |x +3|
+ move left - move
right
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y = log 2 x
y = log 2 (x + 2)
left2
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y = x
y = - x Flip about x
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1
y = e x
y = - e x
flip
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1
y = - ln x
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1
y = - e -x
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Given the graph of y = f(x),
To graph y = f(x) ± a,
Move the graph of y = f(x) up/down a
units
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Given the graph of y = f(x),
To graph y = f(x ± a),
Move the graph of y = f(x) left/rt a units + is left, - is right!
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Given the graph of y = f(x),
To graph y = -f(x),
flip the graph of y = f(x) with respect
to the x-axis.
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The graph of y = - f(x) is
flipped about the x-axis.
The graph of y = f(-x) is
flipped about the y-axis.
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y = -x
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1y = e -
x
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1
y = ln (-x)
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Given the graph of y = f(x),
To graph y = kf(x),
Multiply all the y values of y = f(x) by k. Steeper if k > 1.
Flatter if k < 1
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y = x
y = 2x steeper
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y = |x|
y = 2|x|
y = ½ |x|
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y = x3
y = 2 x 3
double they-values
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y = x3
y = ½ x 3
half they-values
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1
y = e x
y = 2 e x
2
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Assignment Cp. 28/ 47-55,83,93
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SAT Problem of the Day!
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y = |x|
y = -|x - 4| + 3
flip right 4up 3
y = -|x - 4| + 3
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y = |x|
y = -2|x - 4| + 3 flip right 4
up 3
y = -2|x - 4| + 3
steeper
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y = x2
y = -(x - 6)2 + 1
flip right 6 up 1
y = -(x - 6)2 + 1
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SAT Problem of the Day
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(f g)(x) f (g(x))
Given two functions y=f(x) and y=g(x).
“f composed with g of x”
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a) (f g)(x)
Example: Finding composite functions.
Given f(x)=2x+3 and g(x)=cos(x). Find…
f (g(x))
2(cos(x)) 3
b) (g f )(x)
g(f (x))
cos(2x 3)
*Usually, (f g)(x) (g f )(x)
2cos(x) 3
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Example: Finding composite functions.
Given f(x)=2x+3 and g(x)=cos(x).
d) What is (g f )( 2)?(g f )( 2)
g(f ( 2))
g(2 2 3)
g( 1) 0.54
c) What is (f g)( )?(f g)( )
f (g( ))
f (cos( ))
f ( 1) 1
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Try this…
Find two functions f and g such that F(x)= (f g)(x)
a) F(x) 3x 6
2b) F(x) ln(x 4x 4)
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Even vs. Odd
ODD functions are symmetric with respect to the origin.
EVEN functions are symmetric with respect to the y-axis.
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Even or Odd?
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Even or Odd?
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Even or Odd?
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Even or Odd?
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Even or Odd?
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Even or Odd?
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Even and Odd FunctionsEven, odd, or neither test:The function y=f(x) is EVEN if f(-x) = f(x).
The function y=f(x) is ODD if f(-x) = -f(x).
Otherwise, it is neither even nor odd.
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ExampleDetermine whether the function is even,
odd, or neither.a) b) 3f (x) x x g(x) 1 cos(x)
3f ( x) ( x) ( x) 3x x
f (x)
odd
g( x) 1 cos( x) 1 cos(x) * cos is even
g(x)
even
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Assignment E
p. 28-29/ 57 – 70
*Test on Unit P: FunctionsThis Thursday! Review Assignment onlinewww.geocities.com/mskadlac