3. Relation
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- A relation is a correspondence between two sets where each
element in the first set, corresponds to at least one element in the
second set.
8. Vertical Line test
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8
A set of points in the xy-plane is the graph of a function if and only
if every vertical line intersects the graph in at most one point.
9. Finding Domain of a function
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9
π π₯ = π¦
Domain : The set of values of x for each of which the corresponding y is a real
number.
Situations where y will not be real
ππ
π΅πππππππ ππππππ Find the domain of
6
π₯ β 4
q= 0 in
π
π Find the domain of
ππ
πβπ
Involving Logarithmic function
In log π π₯
When a= 0/1/ -ve
When x = 0/-ve
Find the domain of log(π₯β3) 80
10. Finding Domain of a function
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Find the domain of the function,
f(x) =
π₯
π₯2+5π₯+6
1)[0,β)
2)(0, β)
3)(- β, -3] U [-2, β)
4)None of these
Answer: None of these
11. Finding Domain of a function
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Find the domain of the function,
f(x) = log ( 7x β 12 β x2 )
1)[3,4)
2)(3 4)
3)(- β, -3] U [-4, β)
4)None of these
Answer: (3,4)
12. Some common functions - CONSTANT
FUNCTION
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12
f(x) = b, b is any real
number.
16. Some common functions - ABSOLUTE VALUE
FUNCTION
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16
π(π₯) = |π₯|
17. Some common functions - RECIPROCAL
FUNCTION
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π π₯ =
1
π₯
, π€βπππ π₯ β 0
18. Some common functions - GREATEST INTEGER
FUNCTION-(Floor Function)
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f(x) = greatest integer less
than or equal to x.
19. Some common functions β SMALLEST INTEGER
FUNCTION (Ceiling Function)
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f(x) = smallest integer
greater than or equal to x.
20. Odd and Even functions
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Symmetric with
Respect to
On Replacing x
with -x
Odd Function origin
Ζ(-x) = - Ζ(x)
Even Function y-axis Ζ(-x) = Ζ(x)
21. Odd and Even functions
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Adding:
β’ Even + Even = even
β’ Odd + odd = odd
β’ Even + Odd = neither even nor odd (unless one function is zero).
Multiplying:
β’ Even * even = even function.
β’ Odd * odd = even function.
β’The product of an even function and an odd function is an odd
function.
22. Problems
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Determine whether the functions are even,
odd, or neither.
a. Ζ(π₯) = π₯2 β 3
b. π(π₯) = π₯5
+ π₯3
c. β(π₯) = π₯2 β π₯
Answer: Even, Odd and Neither
even nor odd
23. Injective, Surjective and Bijective
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Injective means
every member of "A"
has its own
unique matching
member in "B".
Surjective means
that every "B" has at
least one matching
"A" (maybe more
than one).
Bijective means
both Injective and
Surjective together.
So there is a perfect
"one-to-one
correspondence"
24. Inverse of a function
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- Pre-image and image interchange
Find the inverse of π π₯ = 4π₯ + 3
Answer : πβ1
π₯ =
π₯β3
4
25. Inverse of a function
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Can we have an inverse function for all
functions?
26. Transformations β Vertical and Horizontal Shifts
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Vertical Shift
To Graph Shift the Graph of Ζ(x)
Ζ(π₯) + π c units upward
Ζ π₯ β π c units upward
Horizontal Shift
To Graph Shift the Graph of Ζ(x)
Ζ(π₯ + π) c units to the left
Ζ π₯ β π c units to the right
28. Transformations β Squeezing and Stretching
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In the direction of π
To Graph Shift the Graph of Ζ(x)
Ζ ππ₯ , c > 1 Compress in the direction of x
Ζ ππ₯ , 0 < c < 1 stretch in the direction of x
π π₯ = sin(π₯) π π₯ = sin(2π₯) β π₯ = sin(
1
2
π₯)
30. Transformations β Squeezing and Stretching
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In the direction of π
To Graph Shift the Graph of Ζ(x)
πΖ π₯ , c > 1 Stretch in the direction of y
πΖ π₯ , 0 < c < 1 Compress in the direction of y
π π₯ = sin(π₯) π π₯ = 2sin(π₯) β π₯ =
1
2
sin(π₯)
33. Practise problems
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Let π(π₯) be a function satisfying π(π₯)π(π¦) = π(π₯π¦) for all
real π₯, π¦. If f(2) = 4, then what is the value of π
1
2
?
1. 0
2. ΒΌ
3. Β½
4. 1/3
5. None of these
CAT 2008
Answer: 1/4
36. Practise problems
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Let π(π₯) = max(2π₯ + 1, 3 β 4π₯), where π₯ is any real
number. Then the minimum possible value of π(π₯) is:
1.
1
3
2. Β½
3.
5
3
4.
4
3
5.
2
3
CAT 2006
Answer: Option 3
Editor's Notes
Solution:
As the modulus curve is shifted downwards, the function should be of the form |f(x)| - c. Hence option 1 and 5 are eliminated.
The quadratic equation inside the modulus function has one root >0 and the other <0.
Hence c < 0, option 2 can be ruled out.
Minimum value occurs at x = -b/2a,
In this case βb/2a < 0
-b/2 <0
-b<0
b>0, hence option 4 is eliminated.
Hence option 3
Solution:
6 cases exist
When x<= 0 and y <= 0 , x+y<= 0,
When x<= 0 and y >= 0 , x+y<= 0,
When x<= 0 and y >= 0 , x+y>= 0
When x>= 0 and y <= 0 , x+y<= 0,
When x>= 0 and y <= 0 , x+y>= 0,
When x>= 0 and y >= 0 , x+y>= 0
Solution:
6 cases exist
When x<= 0 and y <= 0 , x+y<= 0,
When x<= 0 and y >= 0 , x+y<= 0,
When x<= 0 and y >= 0 , x+y>= 0
When x>= 0 and y <= 0 , x+y<= 0,
When x>= 0 and y <= 0 , x+y>= 0,
When x>= 0 and y >= 0 , x+y>= 0