M110 CLASS NOTES

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M110 CLASS NOTES

• SECTION 3.7
• One-to-One Functions

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REIEW OF IDENTITIES

• 0 is the identity for the operation
• 3 0 3
• 1 is the identity for the operation ?
• 3?1 3
• What is the identity for the operation ? ?
• f?? f for every function f

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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• f?? f for every function f

Call it I or I(x)
f?I(x) f(x)
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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• f?? f for every function f

f?I(x) f(x) f(I(x)) f(x) If you stare
at this long enough it jumps out at you
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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• f?? f for every function f

f?I(x) f(x) f(I(x)) f(x) If you stare
at this long enough it jumps out at you
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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• f?? f for every function f

f?I(x) f(x) f(I(x)) f(x) If you stare
at this long enough it jumps out at you
I(x) x
But since the operation ? is not commutative, we
must check that it works in the other order as
well.
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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• f?? f for every function f

I(x) x
I?f(x)
But since the operation ? is not commutative, we
must check that it works in the other order as
well.
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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• f?? f for every function f

I(x) x
I?f(x) I(f(x)) f(x)
But since the operation ? is not commutative, we
must check that it works in the other order as
well.
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REIEW OF IDENTITIES

• What is the identity for the operation ? ?
• I(x) x

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REIEW OF INVERSES

• -3 is the inverse for 3 relative to , because 3
  -3 0
• 1/3 is the inverse for 3 relative to ?, because
  3?1/3 1
• If g is going to be an inverse for f, then f?g
  I and g?f I

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REIEW OF INVERSES

• If g is going to be an inverse for f, then f?g
  I and g?f I

f ?g(x) I(x) and g?f(x) I(x)
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REIEW OF INVERSES

• If g is going to be an inverse for f, then f?g
  I and g?f I

f ?g(x) I(x) and g?f(x) I(x)
f(g(x)) x and g(f(x)) x
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DEFINITION

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

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THE DEFINITION IN DIAGRAMS

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

f
x f(x)
g(f(x))
g
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THE DEFINITION IN DIAGRAMS

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

f
f(g(x))
g(x) x
g
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THE DEFINITION IN GRAPHS

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

f(x)
x
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THE DEFINITION IN GRAPHS

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

f(x)
x g(f(x))
x
f(x)
x
The domain values for g are f(x) values
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THE DEFINITION IN GRAPHS

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

f(x)
x g(f(x))
x
f(x)
x
The interchange of x and y values has reflected
the point in the line y x.
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THE DEFINITION IN GRAPHS

• If g is an inverse for f, then
• f(g(x)) x and g(f(x)) x

x
x
If we do this for each ppoint it reflects the
graph in the line y x.
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THE DEFINITION IN GRAPHS

• If g is an inverse for f, then the graph for g is
  a reflection of the graph of x in the line y x.

x
x
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

x
x
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

First, draw the line y x.
x
x
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

Then draw the reflection.
x
x
These two points wont move.
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

Then draw the reflection.
x
x
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

There is only one problem. Do you see it?
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

There is only one problem. Do you see it?
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

THE GRAPH IS NOT THE GRAPH OF A FUNCTION!
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

These two dots that were horizontal
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THE DEFINITION IN GRAPHS

• If the graph of y f(x) is given, find the graph
  of the inverse y g(x).

Reflected to dots that were vertical
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THE HORIZONTAL LINE TEST

• If any horizontal line intersects the graph in
  more than one point then the graph is not the
  graph of an invertible function.

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THE HORIZONTAL LINE TEST

• Which of the following graphs are the graphs of
  invertible functions?

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THE HORIZONTAL LINE TEST

• Which of the following graphs are the graphs of
  invertible functions?

YES
NO NO
YES
YES
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NOTES

• An invertible function is also called a one to
  one function.
• The notation for the inverse of a function f is
  f-1

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NOTES

• Even though
• 3-1 1/3
• x-1 1/x
• a-1 1/a
• f-1 ? 1/f !! (if we are assuming f is a
  function)

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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
STEP 1 Set y f(x) y 5x/(2x
1)
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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
STEP 1 Set y f(x) y 5x/(2x
1) STEP 2 interchange the x and y
x 5y/(2y 1)
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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
STEP 1 Set y f(x) y 5x/(2x
1) STEP 2 Interchange the x and y.
x 5y/(2y 1) STEP 3 Solve for y.
5xy 2y - 1
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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
STEP 1 Set y f(x) y 5x/(2x
1) STEP 2 Interchange the x and y.
x 5y/(2y 1) STEP 3 Solve for y.
5xy 2y 1 5xy 2y -1
y(5x 2) -1
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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
STEP 1 Set y f(x) y 5x/(2x
1) STEP 2 Interchange the x and y.
x 5y/(2y 1) STEP 3 Solve for y.
5xy 2y 1 5xy 2y -1
y(5x 2) -1 y -1/(5x 2)
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EXAMPLE 1
Find f-1(x) if f(x) 5x / (2x-1)
STEP 1 Set y f(x) y 5x/(2x
1) STEP 2 Interchange the x and y.
x 5y/(2y 1) STEP 3 Solve for y.
y -1/(5x 2) STEP 4 Write f-1
f-1(x) -1/(5x 2)
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End of Section 3.7