Transcript y`s - KSU Math Home

This time
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(f+g)(x)=f(x)+g(x)
(f-g)(x)=f(x)-g(x)
(fg)(x)=f(x)*g(x)
(f/g)(x)=f(x)/g(x), g(x)≠0
(f∘g)(x)=f(g(x))
Things to remember
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Function notation
ƒ(x)=2x-1 is a function definition
x is a number
ƒ(x) is a number
2x-1 is a number
ƒ is the action taken to get from x to ƒ(x)
– Multiply by 2 and add -1
Things to remember
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Function notation
ƒ(x)=2x-1 is a function definition
3 is a number
ƒ(3) is a number
2*3-1 is a number (it’s 5)
ƒ is the action taken to get from 3 to ƒ(3)
– Multiply by 2 and add -1
Practice using notation
f
x® f (x)
g
x®g(x)
f
3® f (3)
f
(x - 3)2 ® f ((x - 3)2 )
f
g(x)® f (g(x))
I can do a function to a number
f (x) = 2x -1
g(x) = (x - 3)
g(3) = (3- 3)
g(3) = 0
2
2
ƒ(g(3)) = ƒ(0) = 2 * 0 -1 = -1
ƒ(g(3)) = -1
What’s going on here?
g
f
x®g(x)® f (g(x))
g
f
3®0® -1
g(x) = 0
f (g(x)) = f (0) = -1
Composing functions algebraically
f (x) = 2x -1 is the function definition
f (3) is the number you get when you plug in 3 for x
Anywhere in the definition that you see an x, write (3).
f (3) = 2(3) -1
f (2a) is the number you get when you plug in 2a for x
Anywhere in the definition that you see an x, write (2a).
f (2a) = 2(2a) -1
f ((x - 3)2 ) is the number you get when you plug in (x - 3)2 for x
Anywhere in the definition that you see an x, write ((x - 3)2 ).
f ((x - 3)2 ) = 2((x - 3)2 ) -1
In diagram form
f is the action *2,-1
g is the action -3, ^2
g
f
x®g(x)® f (g(x))
-3,^2
*2,-1
x®(x - 3)2 ®2((x - 3)2 ) -1
WARNING
• Parentheses are ambiguous
• When you have two NUMBERS, a(b) means
“multiply a and b”
• When you have a FUNCTION, a(b) means “do
the action called a to the number b.”
• Always keep track of what’s a function and
what’s a number.
The most common confusion of all time
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(f+g)(x)=f(x)+g(x)
(f-g)(x)=f(x)-g(x)
(fg)(x)=f(x)*g(x)
(f/g)(x)=f(x)/g(x), g(x)≠0
(f∘g)(x)=f(g(x))
COMPARISON
f (x) = 2x -1, g(x) = (x - 3)2
(fg)(3)=f(3)g(3)
(f∘g)(3)=f(g(3))
g(3) = (3- 3)2
g(3) = 0
ƒ(g(3)) = ƒ(0) = 2 * 0 -1 = -1
ƒ(g(3)) = -1
-1≠0
f (3) = 5
g(3) = 0
f (3)g(3) = ( 5) ( 0) = 0
( fg)(3) = 0
WARNING
(fg)(x) and f(g(x)) are not the same thing
– (fg)(x) means “do f to x, then do g to x, then multiply the
numbers f(x) and g(x).”
– f(g(x)) means “do g to x, get the number g(x), then do f to
the number g(x)”
• No multiplying.
In picture form
f(x)
f
x
*
g
f(x)g(x)
g(x)
Is not the same as
g
x
f
g(x)
f(g(x))
Interpretive Dance
All about multiplication
Time to dance!
Add 1 to each number
Add -1 to each number
Multiply each number by 0.5
Multiply each number by 2
Multiplication is not repeated addition
• Addition is shifting
• Multiplication is stretching
– And shrinking
• No amount of repeated shifting will give you a
stretch
Transformations of functions
And their graphs
This is a graph of a function called ƒ
Let g(x)=ƒ(x)+1.5
What does a graph of g look like?
g(3)=ƒ(3)+1.5
(3,f(3)+1.5)
(3,f(3))
g(3)=ƒ(3)+1.5
(3,f(3)+1.5)
(3,f(3))
The x is
still the
same,
But the y
is 1.5
higher
Draw a graph where all the x’s are the same and all the
y’s are 1.5 higher.
(3,f(3)+1.5)
(3,f(3))
The x is
still the
same,
But the y
is 1.5
higher
The graph of f(x)+1.5 is the graph of f(x) shifted up by 1.5
(3,f(3)+1.5)
(3,f(3))
The x is
still the
same,
But the y
is 1.5
higher
Draw a graph where all the x’s are the same and all the
y’s are 1.5 higher.
(3,f(3)+1.5)
(3,f(3))
The x is
still the
same,
But the y
is 1.5
higher
Graphing Transformations
• The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.
This is a graph of a function called ƒ
Let g(x)=ƒ(x-1)
What does a graph of g look like?
g(3)=ƒ(3-1)=ƒ(2)
(2,f(2))
(3,f(2))
g(3)=ƒ(3-1)=ƒ(2)
The y of
g(3) is
the same
as the y
of f(2)
(2,f(2))
(3,f(2))
g(3)=ƒ(3-1)=ƒ(2)
The y of
g(3) is
the same
as the y
of f(2)
(2,f(2))
(3,f(2))
Thinking
from ƒg,
The y is the
same, but
the x
needed to
make that y
is 1 bigger
Draw a graph where the y’s are the same, but
you need a 1 bigger x to make each one.
The y of
g(3) is
the same
as the y
of f(2)
(2,f(2))
(3,f(2))
Thinking
from ƒg,
The y is the
same, but
the x
needed to
make that y
is 1 bigger
Graphing Transformations
• The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.
• The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a.
– NOTE THE MINUS SIGN
This is a graph of a function called ƒ
Let g(x)=2ƒ(x)
g(-1.5)=2ƒ(-1.5)
(-1.5,f(-1.5))
(-1.5,2f(-1.5))
g(-1.5)=2ƒ(-1.5)
The x is still
the same,
But the y is
twice as far
away from
zero
(-1.5,f(-1.5))
(-1.5,2f(-1.5))
Draw a graph where the x’s stay the same, but
the y’s are twice as far away from zero.
The x is still
the same,
But the y is
twice as far
away from
zero
(-1.5,f(-1.5))
(-1.5,2f(-1.5))
Graphing Transformations
• The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.
• The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a.
– NOTE THE MINUS SIGN
• The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r.
This is a graph of a function called ƒ
Let g(x)=ƒ(2x)
g(1.5)=ƒ(2*1.5)=ƒ(3)
(1.5,f(3))
(3,f(3))
The y is still
the same,
but the x
needed to
make that y
is half as big
Draw a graph where the y’s stay the same, but the x’s
needed to make those graphs are half as big.
(1.5,f(3))
(3,f(3))
The y is still
the same,
but the x
needed to
make that y
is half as big
Graphing Transformations
• The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.
• The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a.
– NOTE THE MINUS SIGN
• The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r.
• The graph for ƒ(sx) is the graph of ƒ(x) squished horizontally by s.
Graphing Transformations
• The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.
• The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a.
– NOTE THE MINUS SIGN
• The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r.
• The graph for ƒ(sx) is the graph of ƒ(x) squished horizontally by s.
• Note the difference!
What does it mean to stretch by a fraction?
g(x)=0.5f(x)
What does it mean to stretch by a negative?
Graphing Transformations
• The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.
• The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a.
– NOTE THE MINUS SIGN
• The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r.
– negative r causes the graph to flip vertically.
• The graph for ƒ(sx) is the graph of ƒ(x) squished horizontally by s.
– Negative s causes the graph to flip horizontally
• Note the difference!
Even Function
A function where a horizontal
flip does not change the graph
Even Function
A function where a horizontal
flip does not change the graph
Graph of the function f(x)
Graph of the horizontal flip f(-x)
Even Function
A function where a horizontal
flip does not change the graph
Graph of the function f(x)
Graph of the horizontal flip f(-x)
Even Function
f(x)=f(-x)
Even Function
Example:
f(x)=x2
f(-x)=(-x)2=x2=f(x)
A function where a horizontal
flip does not change the graph
Graph of the function f(x)
Graph of the horizontal flip f(-x)
Even Function
f(x)=f(-x)
Odd Function
A function where a horizontal
flip has same graph as a vertical flip.
Odd Function
A function where a horizontal
flip has same graph as a vertical flip.
Function: f(x)
Horizontal Flip: f(-x)
Vertical Flip: -f(x)
Odd function
f(-x)=-f(x)
Odd Function
Example
f(x)=x3
f(-x)=(-x)3=-(x3)=-f(x)
A function where a horizontal
flip has same graph as a vertical flip.
Function: f(x)
Horizontal Flip: f(-x)
Vertical Flip: -f(x)
Odd function
f(-x)=-f(x)
Write an equation for a function that has the
graph of x2 but is shifted right 3 units and up
4 units.
a)
b)
c)
d)
e)
(x-3)2+4
(x-3)2-4
(x+3)2+4
(x+3)2-4
None of the above
Write an equation for a function that has the
graph of x2 but is shifted right 3 units and up
4 units.
a)
b)
c)
d)
e)
(x-3)2+4
(x-3)2-4
(x+3)2+4
(x+3)2-4
None of the above
Is f(x)=x2+3x-4
a)
b)
c)
d)
Even, not odd
Odd, not even
Both even and odd
Neither even nor odd
Is f(x)=x2+3x-4
a)
b)
c)
d)
Even, not odd
Odd, not even
Both even and odd
Neither even nor odd
f(-x)=(-x)2-3x-4=x2-3x-4
f(-x)≠f(x) NOT EVEN
F(-x)≠-f(x) NOT ODD