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0211 ch 2 day 11
1. 2.8 One-to-One Functions
& Their Inverses
Deuteronomy 7:9 Know therefore that the LORD your God
is God, the faithful God who keeps covenant and steadfast
love with those who love him and keep his commandments,
to a thousand generations.
2. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
3. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
4. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
y = sin(x)
5. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
y = sin(x)
6. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
y = sin(x)
Vertical Line Passes
7. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
y = sin(x)
Vertical Line Passes
8. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
y = sin(x)
Vertical Line Passes
Horizontal Line Fails
9. A one-to-one function is a function where every
input has exactly one output and every output
has exactly one input.
Must pass vertical and horizontal line tests
y = sin(x)
Vertical Line Passes
Horizontal Line Fails
Sine is not one-to-one
13. 1
−1
y = sin (x) y=
x
ArcSin is The Inverse Function
one-to-one is one-to-one
14. If a function is a one-to-one function,
then it’s inverse is also a function.
15. If a function is a one-to-one function,
then it’s inverse is also a function.
Review: How to find the inverse of a function:
16. If a function is a one-to-one function,
then it’s inverse is also a function.
Review: How to find the inverse of a function:
1. interchange x and y
17. If a function is a one-to-one function,
then it’s inverse is also a function.
Review: How to find the inverse of a function:
1. interchange x and y
2. solve for y
21. Example: Find the inverse of y = 3x − 4
x = 3y − 4
x + 4 = 3y
x+4
y=
3
22. Example: Find the inverse of y = 3x − 4
x = 3y − 4
x + 4 = 3y
x+4
y=
3
Graph both ... notice they are symmetric about
the y=x line
x+4
y1 = 3x − 4 y2 =
3
24. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
−1
⎟ − 4
⎝ 3 ⎠
25. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
−1
⎟ − 4
⎝ 3 ⎠
= x+4−4
26. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
−1
⎟ − 4
⎝ 3 ⎠
= x+4−4
=x
27. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
( 3x − 4 ) + 4
−1
⎝ 3 ⎠ ⎟ − 4 f −1
( f ( x )) =
3
= x+4−4
=x
28. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
( 3x − 4 ) + 4
−1
⎝ 3 ⎠ ⎟ − 4 f −1
( f ( x )) =
3
3x
= x+4−4 =
3
=x
29. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
( 3x − 4 ) + 4
−1
⎝ 3 ⎠ ⎟ − 4 f −1
( f ( x )) =
3
3x
= x+4−4 =
3
=x =x
30. Consider the composition of these two
inverse functions:
−1 x+4
f (x) = 3x − 4 f (x) =
3
⎛ x + 4 ⎞
f ( f ( x )) = 3 ⎜
( 3x − 4 ) + 4
−1
⎝ 3 ⎠ ⎟ − 4 f −1
( f ( x )) =
3
3x
= x+4−4 =
3
=x =x
Since inverse functions are symmetric about
the y=x line, it should make sense that the
composition of them would be x.
31. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
32. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
y= 3 x−4
33. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
y= 3 x−4
x = 3 y−4
34. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
y= 3 x−4
x = 3 y−4
3
x = y−4
35. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
y= 3 x−4
x = 3 y−4
3
x = y−4
3
y= x +4
36. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
y= 3 x−4
x = 3 y−4
3
x = y−4
3
y= x +4
37. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
3
y= x−4 f(f −1
( x )) = 3
(x 3
+ 4) − 4
x = 3 y−4
3
x = y−4
3
y= x +4
38. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
3
y= x−4 f(f −1
( x )) = 3
(x 3
+ 4) − 4
x = 3 y−4 = x 3 3
3
x = y−4
3
y= x +4
39. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
3
y= x−4 f(f −1
( x )) = 3
(x 3
+ 4) − 4
x = 3 y−4 = x 3 3
3
x = y−4
=x
3
y= x +4
40. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
3
y= x−4 f(f −1
( x )) = 3
(x 3
+ 4) − 4
x = 3 y−4 = x 3 3
3
x = y−4
=x
3
y= x +4
41. Let f (x) = x − 4 . Find
3
f −1 (x) .
Verify the inverse both algebraically and graphically.
3
y= x−4 f(f −1
( x )) = 3
(x 3
+ 4) − 4
x = 3 y−4 = x 3 3
3
x = y−4
=x
3
y= x +4
Graph and show symmetry about y=x:
3
3
y1 = x − 4 y2 = x + 4 y3 = x
42. HW #10
“We’re not all designed to be straight A students,
celebrities, world-class athletes or the CEO of a
major corporation. But we are designed to make
the most of the skills and abilities we do
possess. Perhaps the most splendid achievement
of all is the continuing quest to surpass
ourselves.” Dennis Waitley
