Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1)...
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Transcript of Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1)...
![Page 1: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/1.jpg)
FunctionsLesson 2
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Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that is perpendicular to the line 4x – 7y = 12.
3. Write the equation of the vertical line that passes through the point (3, 2).
075 yx
4
7m
3x
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Relation Relation – pairs of quantities that are related
to each other
Example: The area A of a circle is related to its radius r by the formula
.2rA
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Function There are different kinds of relations.
When a relation matches each item from one set with exactly one item from a different set the relation is called a function.
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Definition of a Function A function is a relationship between two
variables such that each value of the first variable is paired with exactly one value of the second variable.
The domain is the set of permitted x values.
The range is the set of found values of y. These will be called images.
![Page 6: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/6.jpg)
Let’s take a look at the function that relates the time of day to the temperature.
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Rules to be a Function
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Is it a Function? For each x, there is
only one value of y.
Therefore, it IS a function.
Domain, x Range, y
1 -3.6
2 -3.6
3 4.2
4 4.2
5 10.7
6 12.1
52 52
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Is it a function? Three different y-
values (7, 8, and 10) are paired with one x-value.
Therefore, it is NOT a function
Domain, x Range, y
3 7
3 8
3 10
4 42
10 34
11 18
52 52
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Function? Is it a function? Name the domain and range.
YES. For every x-value, there is only one value of y.
Domain: (3, 4, 5, 7, 8) Range: (-5, -8, 6, 10, 2)
{(3, -5), (4, -8), (5, 6), (7, 10), (8, 2)}
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Function? Is it a function? State the domain and range.
No. The x-value of 5 is paired with two different y-values.
Domain: (5, 6, 3, 4, 12) Range: (8, 7, -1, 2, 9, -2)
{(5, 8), (6, 7), (3, -1), (4, 2), (5, 9), (12, -2)
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Function? Is it a function? Name the domain and range.
Yes. For every x-value, there is only one value of y.
Domain: (-2, 4, 3, 7, 9, 2) Range: (3, 6, 1, -3, 8)
{(-2, 3), (4, 6), (3, 1), (7, 6), (9, -3), (2, 8)}
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Function?
YES
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Vertical Line Test Used to determine if a graph is a function.
If a vertical line intersects the graph at more than one point, then the graph is NOT a function.
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NOT a function
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IS a function
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You Try…...
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You Try….
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You Try: Is it a Function? YES
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You Try…Is it a function? YES.
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You Try…Is it a Function? NO.
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Is it a function? Give the domain and range.
4,4:
2,4:
Range
Domain
FUNCTION
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Give the Domain and Range.
2:
1:
yRange
xDomain
30:
22:
yRange
xDomain
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IB Notation….
When a function is defined for all real values, we write the domain of f as
![Page 25: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/25.jpg)
Functional Notation
We have seen an equation written in the form y = some expression in x.
Another way of writing this is to use functional notation.
For Example, you could write y = x²
as f(x) = x².
![Page 26: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/26.jpg)
![Page 27: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/27.jpg)
Functional Notation
f(x) = 3x + 5
Find:
( 2)f (0)f (5)f
1
56
523
5
50
503
20
515
553
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Functional Notation
Find:
( 3)f (0)f (4)f
2( ) 3 2f x x x
32
230
2327
2333 2
2
200
2003 2
46
2448
24163
2443 2
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Functional Notation
Find:
( )f m
2( ) 2f x x x
3( )f m
22 mm
85
23933
2333
233
2
2
2
mm
mmmm
mmm
mm
![Page 30: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/30.jpg)
Let’s look at Functions Graphically
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Find: 2 4( ) ( )f g
( )f x ( )g x
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Find: 1 0( ) ( )f g
( )f x ( )g x
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Find: 2 1( ) ( )f g
( )f x ( )g x
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Find: 5 0( ) ( )f g
( )f x ( )g x
![Page 35: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/35.jpg)
Find: 4 1( ) ( )f g
( )f x ( )g x
![Page 36: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/36.jpg)
Find: 4 2( ) ( )f g
( )f x ( )g x
![Page 37: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/37.jpg)
Find: 2 0( ) ( )f g
( )f x ( )g x
![Page 38: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/38.jpg)
Find: 5 3( ) ( )g f
( )f x ( )g x
![Page 39: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/39.jpg)
Piecewise-Defined Function
![Page 40: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/40.jpg)
A piecewise-defined function is a function that is defined by two or more equations over a specified domain.
The absolute value function
can be written as a piecewise-defined function.
The basic characteristics of the absolute value function are summarized on the next page.
xxf
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Example
Evaluate the function when x = -1 and 0.
![Page 43: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/43.jpg)
Domain of a Function
![Page 44: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/44.jpg)
The domain of a function can be implied by the expression used to define the function
The implied domain is the set of all real numbers for which the expression is defined.
For example,
![Page 45: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/45.jpg)
The function has an implied
domain that consists of all real x other than
x = ±2
The domain excludes x-values that result in division by zero.
![Page 46: Functions Lesson 2. Warm Up 1. Write an equation of the line that passes through the points (-2, 1) and (3, 2). 2. Find the gradient of the line that.](https://reader035.fdocuments.net/reader035/viewer/2022062802/56649eaa5503460f94baf7c2/html5/thumbnails/46.jpg)
Another common type of implied domain is that used to avoid even roots of negative numbers.
EX:
is defined only for
The domain excludes x-values that result in even roots of negative numbers.
.0x