The document defines one-to-one functions and their inverses. A one-to-one function maps each element in the domain to a unique element in the range. The inverse of a one-to-one function undoes the original mapping. To find the inverse function algebraically, interchange the x and y variables and isolate y. Graphically, flip the graph across the line y=x to find the inverse. Examples are provided of finding inverses both algebraically and graphically. Homework problems involve finding specific inverses and their domains and ranges.

1. Inverse of Functions
Mathematics 4
June 22, 2012
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2. One-to-one Function
Definition
One-to-one function
A function f is one-to-one if for any a, b ∈ {domain of f },
f (a) = f (b).
Its graph passes the horizontal line test.
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3. One-to-one Function
Examples
f (x) = x3
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4. One-to-one Function
Examples
f (x) = x2
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5. One-to-one Function
Examples
f (x) = x2 f (x) = x2 , x ≥ 0
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6. One-to-one Function
Examples
x
f (x) = −3
4
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7. One-to-one Function
Examples
4
f (x) =
x−2
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8. Inverse of a Function
Definition
Inverse of a Function
1 If f is a one-to-one function, and
2 g(f (x)) = x, then:
g(x) is an inverse function of f (x)
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9. Inverse of a Function
Definition
Inverse of a Function
1 If f is a one-to-one function, and
2 g(f (x)) = x, then:
g(x) is an inverse function of f (x)
g(x) = f −1 (x)
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10. Inverse of a Function
Diagrams
f (x)
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11. Inverse of a Function
Diagrams
f (x) f −1 (x)
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12. Inverse of a Function
Diagrams
g(x)
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13. Inverse of a Function
Diagrams
g(x) Not a function
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14. Inverse of a Function
Diagrams
g(x) Not a function
There is no inverse for g(x)
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15. Inverse of a Function
Solving for the inverse
To find the inverse of a function, interchange the x− and y− variables and
isolate y.
Examples:
1 f (x) = x3 + 1
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16. Inverse of a Function
Solving for the inverse
To find the inverse of a function, interchange the x− and y− variables and
isolate y.
Examples:
1 f (x) = x3 + 1
√
2 g(x) = 3 y
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17. Inverse of a Function
Solving for the inverse
To find the inverse of a function, interchange the x− and y− variables and
isolate y.
Examples:
1 f (x) = x3 + 1
√
2 g(x) = 3 y
3 h(x) = 4x − 3
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18. Inverse of a Function
Solving for the inverse
To find the inverse of a function, interchange the x− and y− variables and
isolate y.
Examples:
1 f (x) = x3 + 1
√
2 g(x) = 3 y
3 h(x) = 4x − 3
x+3
4 F (x) =
4x + 5
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19. Inverse of a Function
Solving for the inverse graphically
To find the inverse of a function graphically, flip the graph such that the
x−axis corresponds to the y−axis, and vice versa.
Examples:
f (x) f −1 (x)
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20. Inverse of a Function
Solving for the inverse graphically
To find the inverse of a function graphically, flip the graph such that the
x−axis corresponds to the y−axis, and vice versa.
Examples:
f (x) f −1 (x)
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21. Inverse of a Function
Solving for the inverse graphically
To find the inverse of a function graphically, flip the graph such that the
x−axis corresponds to the y−axis, and vice versa.
Examples:
f (x) f −1 (x)
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22. Inverse of a Function
Homework 8
Find f −1 . Verify that f −1 (f (x) = f (f −1 (x)) = x. Find also the domain
and range of f −1 .
1 f (x) = 5x − 2
2 f (x) = 8 − x3
x−1
3 f (x) =
x+2
4 f (x) = 4 − 3x
5 f (x) = x3 + 1
4
6 f (x) = −
x+1
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