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.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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