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OBJECTIVE:
To be able to find inverses of functions.
RELEVANCE:
To be able to model a set of raw data after a function to best represent that data.
1.
Inverses of Functions
Section 1.6
2.
To be able to
find inverses of
3.
To be able to model
a set of raw data
after a function to
best represent that
5.
Warm Up: Graph & give the
domain & range.
x 5, x 3
f ( x) 2, 3 x 1
x 4, x 1
Answer on Next Slide
6.
Warm Up #3: Graph & give
the domain & range.
x 5, x 3
f ( x) 2, 3 x 1
x 4, x 1
x y x y
-3 2 1 -3
-4 1 2 -2
-5 0 3 -1
4 0
D : , 3 3,
R : ,
7.
Intro to Inverses
Sneetches Video or Read Story Quickly
A function describes the relationship
between 2 variables, applying a rule to
an input that generates exactly one
output.
For such relationships, we are often
compelled to “reverse” or “undo” the
rule.
8.
The Sneetches Story
Some of the Sneetches have stars on their
yellow bellies; some do not.
In the story, Sylvester McMonkey McBean
builds a machine that can apply these stars.
Sneetches without stars pay McBean to make
them look like their start-bellied friends.
When all Sneetches appear the same,
another problem emerges. So………………
9.
The Sneetches Story
McBean builds a second machine to
“reverse” the action of the first one –
one that removes the stars.
Can you identify the math connections
to this story?
10.
Math Connections
functions
McBean’s machine represents____________.
The second machine “reverses” the action of
the first, thereby acting as an ____________.
inverse
11.
Progress of Inverses
Throughout Math
Learned Addition and then its inverse
operation Subtraction.
Learned Multiplication and then its
inverse operation Division.
Learning Perfect Squares connects with
extracting Square Roots
Basically – Inverses are a second
operation that reverses the first
one!
12.
Calculator Inverses
Take a look at your GDC’s and observe
the keys.
Do you notice that inverse operations
of many calculator commands are
“second” functions?
A calculator key pairs an operation or
function with its inverse.
13.
Inverse Function Partner
Share.
List the sequence of steps needed to
evaluate this function:
1. Square the Input
2. Multiply by 9
3. Add 4
14.
Inverse Functions Partner
Divide into groups of 3 or 4.
Each group will be given two functions.
For each function:
1) List the steps of each function
2) Provide 3 ordered pairs that will
satisfy the function.
Seek out other teams who had the other half of
the function share information. For example, a
team that had 2A will seek out a team that had
2B.
Class Discussion – What do you notice about the
“undoing” of inverses of functions?
15.
Warm Up #3: Graph & give
the domain & range.
x 5, x 3
f ( x) 2, 3 x 1
x 4, x 1
Answer on Next Slide
16.
Warm Up #3: Graph & give
the domain & range.
x 5, x 3
f ( x) 2, 3 x 1
x 4, x 1
x y x y
-3 2 1 -3
-4 1 2 -2
-5 0 3 -1
4 0
D : , 3 3,
R : ,
17.
Inverse of a relation
The inverse of the ordered pairs (x, y) is
the set of all ordered pairs (y, x).
The Domain of the function is the range
of the inverse and the Range of the
function is the Domain of the inverse.
1 In other words, switch the
Symbol: f ( x) x’s and y’s!
18.
Example: {(1,2), (2, 4), (3, 6), (4, 8)}
2 ,1 , 4 , 2 , 6 , 3 , 8 , 4
19.
Function notation? What is
really happening when you find
the inverse?
Find the inverse of f(x)=4x-2
x *4 -2 4x-2
(x+2)/4 /4 +2 x
f 1
x x 2
4
20.
To find an inverse…
Switch the x’s and y’s.
Solve for y.
Change to functional notation.
21.
Find Inverse:f ( x ) 8 x 1
f ( x) 8x 1
y 8x 1
x 8 y 1
8 y x 1
x 1
y
8
1 x 1
f x
8
22.
Find Inverse:f ( x ) 8 x 2
f ( x) 8x 2
y 8x 2
x 8 y 2
8 y x 2
x2
y
8
1 x2
f
8
23.
3x 1
Find Inverse:f ( x )
2
3x 1
f x
2
3x 1
y
2
3 y 1
x
2
3 y 1 2 x
3 y 2 x 1
2x 1
y
3
2x 1
f 1
3
24.
2
Find Inverse: f ( x ) x 4
2
f x x 4
2
y x 4
2
x y 4
2
y x 4
y x 4
1
f ( x) x 4
25.
Draw the inverse. Compare to
the line y = x. What do you
notice?
y x
5,3 3, 5
4,2 2, 4
3,1 1, 3
2,1 1, 2
1,1 1, 1
0,1 1,0
1,5 5,1
26.
Graph the inverse of the
following:
The function and
x y its inverse are
symmetric with
0 -5
-3 -4 respect to the
1 2 Line y = x.
1 4
27.
How Does an Inverse Exist
28.
More on One-To-One
Recall that a function is a set of
ordered pairs where every first
coordinate has exactly one second
coordinate.
A one-to-one function has the
added constraint that each 2nd
coordinate has exactly one 1st
coordinate.
32.
Things to note..
1 is the range of f(x).
The domain off x
The graph of an inverse function can be
found by reflecting a function in the line
y=x.
x this
Check 1 by plotting y = 3x + 1 and
y
3 on your graphicTake
calculator.
a look
33.
Reflecting..
5
y=3x+1
4
3 y=x
2
1
y=(x-1)/3
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
-2
-3
-4
-5
34.
Find the inverse of the function.
x y
f ( x) x 2
x y 2
y x 2
Is the inverse also a function? Let’s look at
the graphs. f x x 2
1
f x x
2
If f x x ,
x y 2
x y2
NOTE: Inverse is NOT
y x Inverse
a function!
35.
Horizontal Line Test
Recall that a function passes the
vertical line test.
The graph of a one-to-one function
will pass the horizontal line test. (A
horizontal line passes through the
function in only one place at a
time.)
36.
Is it an Inverse?
A function can only have an inverse
if it is one-to-one.
You can use the horizontal line test
on graphical representations to see
if the function is one-to-one.
37.
Composition and Inverses
If f and g are functions and
( f g )( x) g f x x,
then f and g are inverses of one
another.
38.
Example: Show that the following
are inverses of each other.
1 2
f x 7 x 2 and g x x
7 7
1 2
f g x 7 x 2 g f x 1 7 x 2 2
7 7
7 7
2 2
x 2 2 x
7 7
x x
The composition of each both produce
a value of x; Therefore, they are inverses
of each other.
39.
3
Are f & g inverses? f ( x ) x 4
g ( x) 3 x 4
3
f g x 3
x4 4 g f x 3 x3 4 4
x 4 4 3 x 3
x x
YE S !
40.
You Try….
Show that
1 3
f x 4 x 3 and g ( x ) x
4 4
are inverses of each other.
f g x g f x x
Therefore , they ARE
inverses of each other.
41.
f ( x) 3x 2
Are f & g inverses?
x2
g ( x)
3
x2
3x 2 2
f g x 3 2
g f x
3 3
x 2 2 3x
x 3
x
YE S !
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