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Section 1.1 Functions
A function f from D into R is a rule that assigns to each element x D a unique element y R.
The set D is called the domain of f and the set of image of D under f is called the range of f.
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b.
a.
c.
.x
.y
.z
P Q
f
Is f a function? YES! Domain of f : { a, b, c}
Range of f : {x, y, z}
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b.
a.
c.
.x
.y
.z
P Q
g
Is g a function? YES! Domain of g: { a, b, c}
Range of g: {x, y}
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a.
c.
.x
.y
.z
P Q
h
Is h a function? YES! Domain of h: { a, c}
Range of h: {x, y}
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b.
a.
c.
.x
.y
.z
P Q
i
Is i a function? NO! i is not a function because ……..
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How graph of a function supposed to be?
Vertical line test: A curve in a plane is a graph of a function if no vertical line intersects the curve in more than one point.
Yes! This is a graph of a function because…
No! This is not a graph of a function because…
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Vertical Line Test
• A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once
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Determine whether the following are graphs of functions
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x=t^2-2, y=5-2t, A parabola
-8 -6 -4 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
-10
-5
5
10
15
20
25
x
f(x)
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r(t)=cos (2t)
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
x
f(x)
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r(t)=sin ((8t)/5)
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
x
f(x)
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f(x)=2cos(x)+sin(2x)
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8
-10
-8
-6
-4
-2
2
4
6
8
x
f(x)
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r(t)=cos(4t) A rose with eight leaves
-0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2 1.4
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
x
f(x)
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r(t)=2sqrt(cos(2t)) Lemniscate
-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
x
f(x)
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f(x)=1/(x̂ 2+2x+2)
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
x
f(x)
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r(t)=(e^cos(t))-2(cos(4t))+(sin(t/ 4))^3 Butterfly !!
-3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
-4
-3
-2
-1
1
2
3
4
5
x
f(x)
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f(x)=0.6x+24cos(0.4x)+0.2x
-25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50
-35
-30
-25
-20
-15
-10
-5
5
10
15
20
25
30
35
40
x
f(x)
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x(t)=sin(3t) , y(t)=sin(4t)
-1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2
-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1
1.2
x
f(x)
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x(t)=t +sin(2t) , y(t)=t+sin(3t)
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-8
-6
-4
-2
2
4
6
8
x
f(x)
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Domain & Range
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Linear Functions
Domain of P ?
Range of P ?
{ x (-, + ) } = R
{ y (-, + ) } = R
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Quadratic function
Domain of f : { x (-, + ) } = R
Range of f : { y [0, +) }
f
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f(x)=x̂ 3
-2 -1.5 -1 -0.5 0.5 1 1.5 2
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
x
f(x)
Domain of f ?
Cubic function
{ x (-, + ) } = R
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What can we conclude about the domain of polynomials?
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Domain & range
Domain of f : { x - < x 0 } Range of f : { y 0 y < }
f
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Domain of f ?
Range of f ?
f
Rational Functions
)},0()0,(|{ x
)},0()0,(|{ y
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Rational Functions
Domain of R?
Range of R?
)},2()2,(|{ x
)},0(|{ y
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Question: how to find the domain of a function without looking at the graph?
Ex 1: Find the domain of .9
1)(
2
xxf
Df= All real numbers except -3 & 3 or can be written as Df= { x (-, -3) (-3, 3) (3,+)}.
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Ex 3: Find the domain of .4
5)(
2
xxf
Ex 2: Find the domain of .3)( xxg
Solution :Ex 2 : ),3[
Ex 3 : ),2()2,(
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Even & Odd functions
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even & odd functionsIf a function f satisfies f(-x) = f(x) for every number x in it’s domain, then f is called an even function.
If a function f satisfies f(-x)= -f(x) for every number x in it’s domain, then f is called an odd function.
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Are the following functions even, odd or neither? f(x) = x2, g(x) = x3,
h(x) = - x2 + 4x – 3
Check it by doing some calculation!
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Just by looking at the graph, how can we tell that the function is even,
odd or neither?
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EVEN FUNCTION
NOTICE THAT f(-x) = f(x)f IS SYMMETRY ABOUT Y-AXIS
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ODD FUNCTION
NOTICE THAT f(-x) = -f(x)f IS SYMMETRY ABOUT ORIGIN