Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that...
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Transcript of Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that...
![Page 1: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/1.jpg)
Section 4.1
Inverse Functions
![Page 2: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/2.jpg)
What are Inverse Operations?
Inverse operations are operations that “undo” each other.
ExamplesAddition and Subtraction are inverse
operationsMultiplication and Division are inverse
operations
![Page 3: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/3.jpg)
How to Find Inverse Relation
• Interchange coordinates of each ordered pair in the relation.
• If a relation is defined by an equation, interchange the variables.
![Page 4: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/4.jpg)
Inverses• When we go from an output of a function
back to its input or inputs, we get an inverse relation. When that relation is a function, we have an inverse function.
• Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation.
Example: Consider the relation g given by
g = {(2, 4), (3, 4), (8, 5). Solution: The inverse of the relation is
![Page 5: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/5.jpg)
Inverse Relation
![Page 6: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/6.jpg)
Inverse Relation• If a relation is defined by an equation,
interchanging the variables produces an equation of the inverse relation.
Example: Find an equation for the inverse of the relation:
y = x2 2x.
Solution: We interchange x and y and obtain an equation of the inverse:
• Graphs of a relation and its inverse are always reflections of each other across the line y = x.
![Page 7: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/7.jpg)
Inverse Relation
![Page 8: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/8.jpg)
One-to-One Functions
4
3
2
1
5
6
7
8
f
4
3
2
1
6
8
9
g
f is one-to-one g is not one-to-one
![Page 9: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/9.jpg)
One-to-One Function
A function f is a one-to-one function if,
for two elements a and b from the domain
of f,
a b implies f(a) f(b)
** x-values do not share the same y-value **
![Page 10: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/10.jpg)
Horizontal Line Test
• Use to determine whether a function is one-to-one
• A function is one-to-one if and only if no horizontal line intersects its graph more than once.
![Page 11: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/11.jpg)
Horizontal-Line Test
Graph f(x) = 3x + 4. Example: From the graph at the left, determine whether the function is one-to-one and thus has an inverse that is a function.
Solution: No horizontal line intersects the graph more than once, so the function is one-to-one. It has an inverse that is a function.
f(x) = 3x + 4
![Page 12: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/12.jpg)
Horizontal-Line Test
Graph f(x) = x2 2. Example: From the
graph at the left, determine whether the function is one-to-one and thus has an inverse that is a function.
Solution: There are many horizontal lines that intersect the graph more than once. The inputs 1 and 1 have the same output, 1. Thus the function is not one-to-one. The inverse is not a function.
![Page 13: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/13.jpg)
Why are one-to-one functions important?
One-to-One Functions have
Inverse functions
![Page 14: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/14.jpg)
Inverses of Functions
If the inverse of a function f is also a function, it is named f 1 and read “f-inverse.”
The negative 1 in f 1 is not an exponent. This does not mean the reciprocal of f.
f 1(x) is not equal to1
( )f x
![Page 15: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/15.jpg)
One-to-One Functions
A function f is one-to-one if different inputs have different outputs.
That is, if a b then f(a) f(b).
A function f is one-to-one if when the outputs are the same, the inputs are the same.
That is, if f(a) = f(b) then a = b.
![Page 16: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/16.jpg)
Properties of One-to-One Functions and Inverses
• If a function is one-to-one, then its inverse is a function.
• The domain of a one-to-one function f is the range of the inverse f 1.
• The range of a one-to-one function f is the domain of the inverse f 1.
• A function that is increasing over its domain or is decreasing over its domain is a one-to-one function.
![Page 17: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/17.jpg)
Inverse Functions
x Y = f(x)
A Bf
f-1
Domain of f-1 = range of f
Range of f-1 = domain of f
![Page 18: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/18.jpg)
How to find the Inverse of a One-to-One Function
1. Replace f(x) with y in the equation.
2. Interchange x and y in the equation.
3. Solve this equation for y.
4. Replace y with f-1(x).
Any restrictions on x or y should be considered.Remember: Domain and Range are interchanged
for inverses.
![Page 19: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/19.jpg)
Example
Determine whether the function f(x) = 3x 2 is one-to-one, and if it is, find a formula for f 1(x).
Solution: The graph is that of a line and passes the horizontal-line
test. Thus it is one-to-one and its inverse is a function.
1. Replace f(x) with y:
2. Interchange x and y:
3. Solve for y: 4. Replace y with f 1(x):
![Page 20: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/20.jpg)
Graph of Inverse f-1 function
• The graph of f-1 is obtained by reflecting the graph of f across the line y = x.
• To graph the inverse f-1 function:Interchange the points on the graph of f to obtain the points on the graph of f-1. If (a,b) lies on f, then (b,a) lies on f-1.
![Page 21: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/21.jpg)
Example
Graph f(x) = 3x 2 and f 1(x) = using the same set of axes.
Then compare the two graphs.
2
3
x
![Page 22: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/22.jpg)
Solution
73
42
20
51
f(x) = 3x 2x
24
11
02
15
x
+ 2
3
xf 1(x) =
![Page 23: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/23.jpg)
Inverse Functions and Composition
If a function f is one-to-one, then f 1 is the unique function such that each of the following holds:
for each x in the domain of f, and
for each x in the domain of f 1.
1 1
1 1
( )( ) ( ( ))
( )( ) ( ( ))
f f x f f x x
f f x f f x x
![Page 24: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/24.jpg)
Example Given that f(x) = 7x 2, use composition of
functions to show that f 1(x) = (x + 2)/7.
Solution: 1 1
1
( )( ) ( ( ))
(7 2)
(7 2) 2
77
7
f f x f f x
f x
x
x
x
1 1( )( ) ( ( ))
2( )
72
7( ) 272 2
f f x f f x
xf
x
x
x
![Page 25: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/25.jpg)
Restricting a Domain
• When the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function.
• In such cases, it is convenient to consider “part” of the function by restricting the domain of f(x). If the domain is restricted, then its inverse is a function.
![Page 26: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/26.jpg)
Restricting the Domain
Recall that if a function is not one-to-one, then its inverse will not be a function.
![Page 27: Section 4.1 Inverse Functions. What are Inverse Operations? Inverse operations are operations that “undo” each other. Examples Addition and Subtraction.](https://reader036.fdocuments.net/reader036/viewer/2022070414/5697c0121a28abf838ccc4de/html5/thumbnails/27.jpg)
Restricting the Domain
If we restrict the domain values of f(x) to those greater than or equal to zero, we see that f(x) is now one-to-one and its inverse is now a function.