Download - Warmup Alg 31 Jan 2012
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Warmup Alg 31 Jan 2012
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Agenda• Don't forget about resources on
mrwaddell.net• Section 6.4: Inverses of functions
• Using “Composition” to prove inverse• Find the inverse of a function or relation
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Practice from last class period’s assignment.
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Section 6.4: Inverses of Functions
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Vocabulary
Domain
Range
Inverse
Composition
The x values of the points
The y values of the points
A Function “flipped”
Putting one function inside another
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Composition of Functions
If f(x) = 5x2 – 2x and g(x) = 4x
Then f(g(x)) is:
f(g(x)) = 5( )2 – 2( ) g(x) g(x)
f(g(x)) = 5( )2 – 2( ) 4x 4x
f(g(x)) = 5(16x2 ) – 2( 4x )
f(g(x)) = 80x2 – 8x
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Composition of Functions 2
If f(x) = 5x2 – 2x and g(x) = 4x
Then g(f(x)) is:
g(f(x)) = 4( ) f(x)
g(f(x)) = 4( ) 5x2 – 2x
g(f(x)) = 20x2 – 8x
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Composition of functions 3
If f(x)=2x and g(x) = 2x2+2 and h(x)= -4x + 3
Find g(h(2))
g(h(2)) = 2( )2 + 2 h(2)
g(h(2)) = 2( )2 + 2 -4(2) + 3
g(h(2)) = 2( )2 + 2 -8 + 3
g(h(2)) = 2( -5 )2 + 2 = 52
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Composition of functions 3
If f(x)=2x and g(x) = 2x2+2 and h(x)= -4x + 3
Find h(g(2))
h(g(2)) = -4( ) + 3 g(2)
h(g(2)) = -4( ) + 3 2(2)2 +2
h(g(2)) = -4( ) + 3 2(4)+2
h(g(2)) = -4( 10 )+ 3 = -37
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What we are doing
The inverse “flips” the picture over!
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Inverse # 2
Find an equation for the inverse of:y = 2x + 3
First, switch the x and ySecond, solve for y.
x = 2y + 3-2y -2y
-2y +x = + 3 -x -x
-2y = -x + 3-2 -2 -2
y = ½ x – 3/2 That’s all there is to it.
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Inverse #2
Now you try. Find the inverse of:
y = - ½x + 3
The inverse is: y = -2x + 6
Prove it! Here’s how.
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Verifying an inverse is true.f(x) = - ½x + 3 and the inverse is: g(x) = -2x + 6
f(g(x)) = -½( ) + 3
f(g(x)) = -½( -2x + 6 ) + 3
f(g(x)) = x - 3 + 3
f(g(x)) = x
g(f(x)) = -2( ) + 6
g(f(x)) = -2(-½ x + 3) + 6
g(f(x)) = x - 6 + 6
g(f(x)) = x
Do (f◦g) and (g◦ f) and if they both equal “x” then they are inverses!
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Non-linear inverse functions
The dashed line is the equation:
y = x
Notice the symmetry in the red and blue graphs!
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Non-linear inversesThe dashed line is the equation:
y = x
Notice the symmetry in the red and blue graphs!
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Checking Inverses #2
Can you show that
y = 2x + 3 and
y = ½ x – 3/2 are inverses of each other?
Do f(g(x)) and g(f(x)) and if they both equal “x” then they are inverses!
Hint: Call the first one “f(x)” and the second one “g(x)” and lose the “y’s”
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Assignment
Section 6.4: 6-11,
15-20,
42-43
Do All, and pick 1 from each group to write complete explanation.