The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
3. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
4. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
e.g. y x 3 x x y 3 y
5. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
e.g. y x 3 x x y 3 y
If in this new relation, for each x value in the domain there is a
maximum of one y value, (i.e. it is a function), then it is called the
inverse function to y = f(x) and is symbolised y f 1 x
6. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
e.g. y x 3 x x y 3 y
If in this new relation, for each x value in the domain there is a
maximum of one y value, (i.e. it is a function), then it is called the
inverse function to y = f(x) and is symbolised y f 1 x
A function and its inverse function are reflections of each other in
the line y = x.
7. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
e.g. y x 3 x x y 3 y
If in this new relation, for each x value in the domain there is a
maximum of one y value, (i.e. it is a function), then it is called the
inverse function to y = f(x) and is symbolised y f 1 x
A function and its inverse function are reflections of each other in
the line y = x.
If a, b is a point on y f x , then b, a is a point on y f 1 x
8. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
e.g. y x 3 x x y 3 y
If in this new relation, for each x value in the domain there is a
maximum of one y value, (i.e. it is a function), then it is called the
inverse function to y = f(x) and is symbolised y f 1 x
A function and its inverse function are reflections of each other in
the line y = x.
If a, b is a point on y f x , then b, a is a point on y f 1 x
The domain of y f x is the range of y f 1 x
9. Inverse Functions
If y = f(x) is a function, then for each x in the domain, there is a
maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
e.g. y x 3 x x y 3 y
If in this new relation, for each x value in the domain there is a
maximum of one y value, (i.e. it is a function), then it is called the
inverse function to y = f(x) and is symbolised y f 1 x
A function and its inverse function are reflections of each other in
the line y = x.
If a, b is a point on y f x , then b, a is a point on y f 1 x
The domain of y f x is the range of y f 1 x
The range of y f x is the domain of y f 1 x
12. Testing For Inverse Functions
(1) Use a horizontal line test
e.g.
i y x 2 y
x
13. Testing For Inverse Functions
(1) Use a horizontal line test
e.g.
i y x 2 y
x
Only has an inverse relation
14. Testing For Inverse Functions
(1) Use a horizontal line test
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation
15. Testing For Inverse Functions
(1) Use a horizontal line test
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation Has an inverse function
16. Testing For Inverse Functions
(1) Use a horizontal line test
OR
2 When x f y is rewritten as y g x , y g x is unique.
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation Has an inverse function
17. Testing For Inverse Functions
(1) Use a horizontal line test
OR
2 When x f y is rewritten as y g x , y g x is unique.
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation Has an inverse function
OR
x y2
18. Testing For Inverse Functions
(1) Use a horizontal line test
OR
2 When x f y is rewritten as y g x , y g x is unique.
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation Has an inverse function
OR
x y2
y x
NOT UNIQUE
19. Testing For Inverse Functions
(1) Use a horizontal line test
OR
2 When x f y is rewritten as y g x , y g x is unique.
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation Has an inverse function
OR OR
x y2 x y3
y x
NOT UNIQUE
20. Testing For Inverse Functions
(1) Use a horizontal line test
OR
2 When x f y is rewritten as y g x , y g x is unique.
e.g.
i y x 2 y ii y x 3 y
x x
Only has an inverse relation Has an inverse function
OR OR
x y2 x y3
y x y3 x
NOT UNIQUE UNIQUE
21. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
22. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x
23. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
24. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1
f x
3 2x
25. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1
f x
3 2x
2x 1 2 y 1
y x
3 2x 3 2y
26. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1
f x
3 2x
2x 1 2 y 1
y x
3 2x 3 2y
3 2 y x 2 y 1
3 x 2 xy 2 y 1
2 x 2 y 3 x 1
3x 1
y
2x 2
27. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1 2x 1
f x 3 1
3 2x 3 2x
f 1 f x
2x 1
2x 1 2 y 1 2 2
y x 3 2x
3 2x 3 2y
3 2 y x 2 y 1
3 x 2 xy 2 y 1
2 x 2 y 3 x 1
3x 1
y
2x 2
28. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1 2x 1
f x 3 1
3 2x 3 2x
f 1 f x
2x 1
2x 1 2 y 1 2 2
y x 3 2x
3 2x 3 2y
3 2 y x 2 y 1 6x 3 3 2x
3 x 2 xy 2 y 1 4x 2 6 4x
2 x 2 y 3 x 1
8x
3x 1 8
y x
2x 2
29. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1 2x 1 3x 1
f x 3 1 2 1
3 2x
f f 1 x
3 2x 2x 2
f 1 f x
2x 1 3x 1
2x 1 2 y 1 2 2 3 2
y x 3 2x 2x 2
3 2x 3 2y
3 2 y x 2 y 1 6x 3 3 2x
3 x 2 xy 2 y 1 4x 2 6 4x
2 x 2 y 3 x 1
8x
3x 1 8
y x
2x 2
30. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. 2x 1 2x 1 3x 1
f x 3 1 2 1
3 2x
f f 1 x
3 2x 2x 2
f 1 f x
2x 1 3x 1
2x 1 2 y 1 2 2 3 2
y x 3 2x 2x 2
3 2x 3 2y
3 2 y x 2 y 1 6x 3 3 2x 6x 2 2x 2
3 x 2 xy 2 y 1 4x 2 6 4x 6x 6 6x 2
2 x 2 y 3 x 1
8x
8x
3x 1 8 8
y x x
2x 2
32. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
33. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
34. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
e.g. i y x 3
y
y x3
x
35. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
e.g. i y x 3
y
Domain: all real x
y x3
Range: all real y
x
36. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
e.g. i y x 3
y
Domain: all real x
y x3
Range: all real y
f 1 : x y 3 x
1
y x 3
37. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
e.g. i y x 3
y
Domain: all real x
y x3
Range: all real y
f 1 : x y 3 x
1
y x 3
Domain: all real x
Range: all real y
38. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
e.g. i y x 3
y
Domain: all real x
y x3
Range: all real y
f 1 : x y 3 x
1
y x 3
Domain: all real x
Range: all real y
39. Restricting The Domain
If a function does not have an inverse, we can obtain an inverse
function by restricting the domain of the original function.
When restricting the domain you need to capture as much of the
range as possible.
e.g. i y x 3
y
Domain: all real x
y x3
Range: all real y
f 1 : x y 3 x
1
1
y x 3
yx 3
Domain: all real x
Range: all real y
47. iii y x 2 y x2 y
Domain: all real x
Range: y 0
x
48. iii y x 2 y x2 y
Domain: all real x
Range: y 0
NO INVERSE x
49. iii y x 2 y x2 y
Domain: all real x
Range: y 0
NO INVERSE x
Restricted Domain: x 0
50. iii y x 2 y x2 y
Domain: all real x
Range: y 0
NO INVERSE x
Restricted Domain: x 0
Range: y 0
51. iii y x 2 y x2 y
Domain: all real x
Range: y 0
NO INVERSE x
Restricted Domain: x 0
Range: y 0
f 1 : x y 2
1
y x 2
52. iii y x 2 y x2 y
Domain: all real x
Range: y 0
NO INVERSE x
Restricted Domain: x 0
Range: y 0
f 1 : x y 2
1
y x 2
Domain: x 0
Range: y 0
53. iii y x 2 y x2 y
Domain: all real x
Range: y 0
NO INVERSE x
Restricted Domain: x 0
Range: y 0
f 1 : x y 2
1
y x 2
Domain: x 0
Range: y 0
54. iii y x 2 y x2 1
y
Domain: all real x yx 2
Range: y 0
NO INVERSE x
Restricted Domain: x 0
Range: y 0
f 1 : x y 2
1
y x 2
Domain: x 0
Range: y 0
55. iii y x 2 y x2 1
y
Domain: all real x yx 2
Range: y 0
NO INVERSE x
Restricted Domain: x 0
Range: y 0
f 1 : x y 2
1
y x 2
Domain: x 0 Book 2
Exercise 1A; 2, 4bdf, 7, 9, 13, 14, 16, 19
Range: y 0