This document provides an overview of key concepts related to functions and graphs including: - The definitions of relations and functions and the differences between them - Common types of functions such as constant, identity, and absolute value functions - Transformations of graphs including vertical and horizontal shifts, squeezing and stretching - Properties of even, odd, and inverse functions - Techniques for finding the domain of a function - Examples of problems involving functions and their solutions

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Functions and Graphs - Concepts

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Relations and Functions

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.

4. Relation – Domain, co-domain and Range
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5. Function
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- A relation in which each element in the domain corresponds to
exactly one element in the range.

6. Function – image and pre-
image
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𝑓 𝑥 = 𝑦
Image = y
Pre-image = x

7. Relation Vs Function
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Relation
Each input has at least one
output
Function
Each input has EXACTLY one output

8. Vertical Line test
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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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𝑓 𝑥 = 𝑦
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
𝟐𝒏
𝑵𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 Find the domain of
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𝑥 − 4
q= 0 in
𝒑
𝒒 Find the domain of
𝟏𝟓
𝒙−𝟔
Involving Logarithmic function
In log 𝑎 𝑥
When a= 0/1/ -ve
When x = 0/-ve
Find the domain of log(𝑥−3) 80

10. Finding Domain of a function
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Find the domain of the function,
f(x) =
𝑥
𝑥2+5𝑥+6
1)[0,∞)
2)(0, ∞)
3)(- ∞, -3] U [-2, ∞)
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 – 12 – x2 )
1)[3,4)
2)(3 4)
3)(- ∞, -3] U [-4, ∞)
4)None of these
Answer: (3,4)

12. Some common functions - CONSTANT
FUNCTION
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f(x) = b, b is any real
number.

13. Some common functions - IDENTITY FUNCTION
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ƒ(𝑥) = 𝑥

14. Some common functions - SQUARE FUNCTION
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𝑓(𝑥) = 𝑥2

15. Some common functions - CUBE FUNCTION
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𝑓(𝑥) = 𝑥3

16. Some common functions - ABSOLUTE VALUE
FUNCTION
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𝑓(𝑥) = |𝑥|

17. Some common functions - RECIPROCAL
FUNCTION
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𝑓 𝑥 =
1
𝑥
, 𝑤ℎ𝑒𝑟𝑒 𝑥 ≠ 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 – 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
ƒ(-x) = - ƒ(x)
Even Function y-axis ƒ(-x) = ƒ(x)

21. Odd and Even functions
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Adding:
• Even + Even = even
• Odd + odd = odd
• Even + Odd = neither even nor odd (unless one function is zero).
Multiplying:
• Even * even = even function.
• Odd * odd = even function.
•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. ƒ(𝑥) = 𝑥2 − 3
b. 𝑔(𝑥) = 𝑥5
+ 𝑥3
c. ℎ(𝑥) = 𝑥2 − 𝑥
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 𝑓 𝑥 = 4𝑥 + 3
Answer : 𝑓−1
𝑥 =
𝑥−3
4

25. Inverse of a function
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Can we have an inverse function for all
functions?

26. Transformations – Vertical and Horizontal Shifts
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Vertical Shift
To Graph Shift the Graph of ƒ(x)
ƒ(𝑥) + 𝑐 c units upward
ƒ 𝑥 − 𝑐 c units upward
Horizontal Shift
To Graph Shift the Graph of ƒ(x)
ƒ(𝑥 + 𝑐) c units to the left
ƒ 𝑥 − 𝑐 c units to the right

27. Transformations – Squeezing and Stretching
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𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = sin(2𝑥) ℎ 𝑥 = sin(
1
2
𝑥)

28. Transformations – Squeezing and Stretching
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In the direction of 𝒙
To Graph Shift the Graph of ƒ(x)
ƒ 𝑐𝑥 , c > 1 Compress in the direction of x
ƒ 𝑐𝑥 , 0 < c < 1 stretch in the direction of x
𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = sin(2𝑥) ℎ 𝑥 = sin(
1
2
𝑥)

29. Transformations – Squeezing and Stretching
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𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = 2sin(𝑥) ℎ 𝑥 =
1
2
sin(𝑥)

30. Transformations – Squeezing and Stretching
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In the direction of 𝒚
To Graph Shift the Graph of ƒ(x)
𝑐ƒ 𝑥 , c > 1 Stretch in the direction of y
𝑐ƒ 𝑥 , 0 < c < 1 Compress in the direction of y
𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = 2sin(𝑥) ℎ 𝑥 =
1
2
sin(𝑥)

31. Transformations – Reflections
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Reflection along coordinate axis
To Graph Shift the Graph of ƒ(x)
−ƒ(𝑥) Reflect along x axis
ƒ −𝑥 Reflect along y axis

32. Practise problems
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Identify the option that could represent the following graph
1. 𝑥2
+ 3𝑥 − 2 + 3
2. 𝑥2 + 3𝑥 + 2 − 3
3. 𝑥2 + 3𝑥 − 2 − 3
4. 𝑥2 − 3𝑥 − 2 − 3
5. 𝑥2 + 3𝑥 − 2 + 3
Answer: Option 3

33. Practise problems
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Let 𝑓(𝑥) be a function satisfying 𝑓(𝑥)𝑓(𝑦) = 𝑓(𝑥𝑦) for all
real 𝑥, 𝑦. If f(2) = 4, then what is the value of 𝑓
1
2
?
1. 0
2. ¼
3. ½
4. 1/3
5. None of these
CAT 2008
Answer: 1/4

34. Practise problems
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If 𝑎1 = 1and 𝑎 𝑛+1 = 2𝑎 𝑛 + 5, 𝑛 = 1,2 … , then 𝑎100 is
equal to
1. (5 × 299 – 6)
2. (5 × 299 + 6)
3. (6 × 299 + 5)
4. (6 × 299 − 5)
CAT 2000
Answer: Option 4

35. Practise problems
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All points in the region R satisfy the inequality 𝑥 + 𝑦 +
𝑥 + 𝑦 ≤ 40. Find the area of region R
Answer: 600

36. Practise problems
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Let 𝑓(𝑥) = max(2𝑥 + 1, 3 − 4𝑥), where 𝑥 is any real
number. Then the minimum possible value of 𝑓(𝑥) is:
1.
1
3
2. ½
3.
5
3
4.
4
3
5.
2
3
CAT 2006
Answer: Option 3