section 1.2 basics of functions. definition of a relation a relation can be expressed as a set of...
TRANSCRIPT
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Section 1.2Basics of Functions
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Definition of a Relation
A relation can be expressed as a set of ordered pairs. The domain of a relation is the set of first elements
in the ordered pairs, and the range is the set of second elements.
Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}
Math 112 Section 1.2
Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}
Domain: {0, 2, 3}
Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}
Domain: {0, 2, 3}
Range: {5, 1, 1, 8}
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Example
Find the domain and the range.
98.6, Felicia , 98.3,Gabriella , 99.1, Lakeshia
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Definition of a Function
A function is a relation for which each element of the domain corresponds to exactly one element of the range.
Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}Function: {(0 , 2), (1 , 8), (5 , 2), (1 , 3)}
In other words, no x coordinate can be paired with more than one y coordinate.
Math 112 Section 1.2
0 52 13 1
8
0 2 1 8 5 3 1
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Example
Determine whether each relation is a function?
1,8 , 2,9 , 3,10
2,3 , 2,4 , 2,5
3,6 , 4,6 , 5,6
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Function Notation
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Function Notation
A function can also be expressed as an equation.
f(x) = x2 + 5x 2
f(3) = 32 + 5(3) 2 = 22
f(1) = (1)2 + 5(1) 2 = 6f(z+2) = (z+2)2 + 5(z+2) 2
= z2 + 4z + 4 + 5z + 10 2= z2 + 9z + 12
Math 112 Section 1.2
“f of x”
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Example
Evaluate each of the following.
2
2
Find f(3) for f(x)=2x 4
Find f(-2) for f(x)=9-x
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Example
Evaluate each of the following.
2
2
Find f(x+2) for f(x)=x 2 4 ?
Is this is same as f(x) + f(2) for f(x)=x 2 4
x
x
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Example
Evaluate each of the following.
2
2
Find f(-x) for f(x)=x 2 4
Is this is same as -f(x) for f(x)=x 2 4?
x
x
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Graphs of Functions
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The graph of a function is the graph of its ordered pairs.
First find the ordered pairs, then graph the functions.
Graph the functions f(x)=-2x; g(x)=-2x+3
x f(x)=-2x (x,y) g(x)=-2x+3 (x,y)
-2 f(-2)=4 (-2,4) g(-2)=7 (-2,7)
-1 f(-1)=2 (-1,2) g(-1)=5 (-1,5)
0 f(0)=0 (0,0) g(0)=3 (0,3)
1 f(1)=-2 (1,-2) g(1)=1 (1,1)
2 f(2)=-4 (2,-4) g(2)=-1 (2,-1)
See the next slide.
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x
y
x
y
f(x)g(x)
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Example
Graph the following functions f(x)=3x-1 and g(x)=3x
x
y
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The Vertical Line Test
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x
y
The first graph is a function, the second is not.
x
y
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x
y
Example
Use the vertical line test to identify graphs in which y is a function of x.
x
y
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Example
Use the vertical line test to identify graphs in which y is a function of x.
x
y
x
y
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Obtaining Information
from Graphs
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Example
Analyze the graph.2( ) 3 4
a. Is this a function?
b. Find f(4)
c. Find f(1)
d. For what value of x is f(x)=-4
f x x x
x
y
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Identifying Domain and Range from a Function’s Graph
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The domain of a function is the set of all x values for which the function is defined.
Domain
x2 4 0
x 2, 2( , 2) (2 , 2) (2 , )
4x
2xf(x)
2
62x f(x)
Domain
2x + 6 0
2x 6x 3[3 , )
Math 112 Section 1.2
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Finding the Domain & Range of a Function
The domain of a function is the set of all x values from the graph.
The range of a function is the set of all y values from the graph.
Domain: ( , )
Range: [1 , )
Math 112 Section 1.2
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x
yIdentify the function's domain and range from the graph
Domain (-1,4]
Range [1,3)
Domain [3, )
Range [0, )
x
y
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Example
Identify the Domain and Range from the graph.
x
y
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Example
Identify the Domain and Range from the graph.
x
y
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Example
Identify the Domain and Range from the graph.
x
y
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Identifying Intercepts
from a Function’s Graph
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Example
Find the x intercept(s). Find f(-4)
x
y
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Example
Find the y intercept. Find f(2)
x
y
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x
y
Example
Find the x and y intercepts. Find f(5).
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Find f(7).
x
y
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Find the Domain and Range.
x
y
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22 3( ) Find f(-1)
7
xf x
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Example
Determine whether each equation defines y as a function of x.
2
2 2
4 8
2 10
16
x y
x y
x y