1. Functions
Mathematics 4
June 20, 2012
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2. Definitions
Relations
Relations
A set of ordered pairs (x, y) such that for each x-value, there corresponds
at least one y-value.
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3. Definitions
Function
Function
A set of ordered pairs (x, y) such that for each x-value, there corresponds
exactly one y-value.
Function
A correspondence from a set X ⊆ R to a set Y ⊆ R where x ∈ X and
y ∈ Y , and y is unique for a specific value of x.
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4. Definitions
One-to-one Function
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5. Definitions
Domain and Range
Domain
The domain of a function is the set of all possible values of x
(independent variable, abscissa) for a given relation or function.
Range
The range of a function is the set of all possible values of y (dependent
variable, ordinate) for a given relation or function.
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6. Examples
Definitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domain
and range.
1 y = x2 + 6x + 4
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7. Examples
Definitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domain
and range.
1 y = x2 + 6x + 4
2 x2 + y 2 = 1
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8. Examples
Definitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domain
and range.
1 y = x2 + 6x + 4
2 x2 + y 2 = 1
1
3 y=
x+1
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9. Examples
Definitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domain
and range.
1 y = x2 + 6x + 4
2 x2 + y 2 = 1
1
3 y=
x+1
1
4 y=
x2 +1
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10. Homework 5
Identify if the following relations are functions, and give the domain and range.
3
1 y=
x+1
3x2 + 1
2 y=
x2 + 2
√
3 y = −x2 + 25
4 x2 + y 2 = 100
5 y + 3 = (x + 4)2
2
6 y=
|x|
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11. Function Notation
Given the equation y = 2x2 + 5
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12. Function Notation
Given the equation y = 2x2 + 5
Using the set-builder notation and the definition of functions:
f = {(x, y) y = 2x2 + 5 }
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13. Function Notation
Given the equation y = 2x2 + 5
Using the set-builder notation and the definition of functions:
f = {(x, y) y = 2x2 + 5 }
From this notation we can use the shorthand:
f (x) = 2x2 + 5
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14. Definitions
Graphs of Functions
The graph of a function
The graph of a function is the set of ALL POINTS in R2 for which
(x, y) ∈ a given function.
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15. Definitions
Graphs of Functions
The graph of a function
The graph of a function is the set of ALL POINTS in R2 for which
(x, y) ∈ a given function.
Vertical Line Test
The graph of a function can be intersected by a vertical line in at most
one point.
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16. Example:
Square root functions
√
Find the graph of the function f = {(x, y) y = 4 − x }:
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17. Example:
Square root functions
√
Find the graph of the function f = {(x, y) y = x − 1 }:
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18. Example:
Absolute value functions
Find the graph of the function f = {(x, y) |y = |x − 3| }:
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19. Homework 6
Sketch the graph and determine domain and range for each function below.
√
1 f = {(x, y) | y = 16 − x2 }
2 g = {(x, y) | y = (x − 1)3 }
x2 − 4x + 3
3 h= (x, y) | y =
x−1
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20. Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
f (x + h) − f (x)
1 if f (x) = 3x2 − 2x + 4
h
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21. Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
f (x + h) − f (x)
1 if f (x) = 3x2 − 2x + 4
h
2 f (−x) if f (x) = 3x4 − 2x2 + 7
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22. Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
f (x + h) − f (x)
1 if f (x) = 3x2 − 2x + 4
h
2 f (−x) if f (x) = 3x4 − 2x2 + 7
3 g(−x) if g(x) = 3x5 − 4x3 − 9x
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23. Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
f (x + h) − f (x)
1 if f (x) = 3x2 − 2x + 4
h
2 f (−x) if f (x) = 3x4 − 2x2 + 7
3 g(−x) if g(x) = 3x5 − 4x3 − 9x
1+x
4 f (x2 − 1) if f (x) =
2x − 1
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24. Operations on Functions
The following notations are indicate an operation between two functions:
(f + g)(x) = f (x) + g(x)
(f − g)(x) = f (x) − g(x)
(f · g)(x) = f (x) · g(x)
f f (x)
(x) =
g g(x)
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25. Operations on Functions
Determine the result of the following function operations:
x+3
1 (f + g)(x) if f (x) = and g(x) = x − 2
x+2
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26. Operations on Functions
Determine the result of the following function operations:
x+3
1 (f + g)(x) if f (x) = and g(x) = x − 2
x+2
f √ √
2 (x) if f (x) = x3 − x2 − 5x − 3 and g(x) = x−3
g
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27. Composition of Functions
Definition
Composition of Functions
Evaluating a function f (x) with another function g(x).
f (g(x)) = (f ◦ g)(x)
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28. Composition of Functions
Evaluate the following composite functions:
x+1 1
1 f (x) = , g(x) = , find (f ◦ g) and (g ◦ f )
x−1 x
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29. Composition of Functions
Evaluate the following composite functions:
x+1 1
1 f (x) = , g(x) = , find (f ◦ g) and (g ◦ f )
x−1 x
√ √
2 f (x) = x2 − 1, g(x) = x − 1, find (f ◦ g)(x) and (g ◦ f )
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30. Odd and Even Functions
Definitions
Even Function
A function f is even if f (−x) = f (x).
Example
1 f (x) = 3x6 − 2x4 + 4x2 + 2
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31. Odd and Even Functions
Definitions
Even Function
A function f is even if f (−x) = f (x).
Example
1 f (x) = 3x6 − 2x4 + 4x2 + 2
2 g(x) = |x| + 2
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32. Odd and Even Functions
Definitions
Odd Function
A function f is odd if f (−x) = −f (x).
Example
1 f (x) = 2x5 − 4x3 + 5x
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33. Odd and Even Functions
Definitions
Odd Function
A function f is odd if f (−x) = −f (x).
Example
1 f (x) = 2x5 − 4x3 + 5x
1
2 g(x) =
x
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34. Odd and Even Functions
Determine if the following functions are odd, even, or neither
(1) (2)
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35. Odd and Even Functions
Determine if the following functions are odd, even, or neither
(3) (4)
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36. Odd and Even Functions
Determine if the following functions are odd, even, or neither
(5) (6)
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37. Odd and Even Functions
Determine if the following functions are odd, even, or neither
(7) (8)
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38. Odd and Even Functions
Symmetry properties
Even functions
The graph of even functions are symmetric with respect to the y-axis.
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39. Odd and Even Functions
Symmetry properties
Even functions
The graph of even functions are symmetric with respect to the y-axis.
Odd functions
The graph of odd functions are symmetric with respect to the origin.
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40. Homework 7
Determine if the function is odd/even/neither, then find the domain, range, and the
graph of the function.
x
1 f (x) =
x2 −4
2 (f + g)(x) if f (x) = x2 + 1 and g(x) = |x|
3 (f − g)(x) if f (x) = |x| + 1 and g(x) = x2 − 2
4 (f · g) if f (x) = x and g(x) = x3
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