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1. Limits and Continuity – Intuitive
Approach Chapter 8
Paper 4: Quantitative Aptitude- Mathematices
Ms. Ritu Gupta B.A. (Hons.) Maths and MA (Maths)

2. Introduction to Function
• Fundamental Knowledge
• Its application
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3. Definition of Function
A function is a term used to define relation between
variables.
A variable y is called a function of a variable x if
for every value of x there is a definite value of y.
Symbolically y = f(x)
We can assign values of x arbitrarily. So x is called
independent variable whereas y is called the dependent
variable as its values depend upon the value of x.
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4. Types of Functions
1. Even Function – A function f(x) is said to be even
function if
f(-x) = f(x)
e.g. f(x) = 2x2 + 4x4
f(-x) = 2(-x)2 + 4(-x)4
= 2x2 + 4x4 = f(x)
Hence 2x2 + 4x4 is an even function.
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5. Types of Functions - Continued
2. Odd Function – A function is said to be odd function if
f(-x) = - f(x)
e.g. f(x) = 3x + 2x5
f(-x) = 3(-x) + 2(-x)5
= -3x - 2 x5
= - (3x + 2 x5) = - f(x)
Hence 3x + 2 x5 is an odd function.
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6. Types of Functions - Continued
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7. Types of Functions - Continued
4. Composite Function – If y = f(x) and x = g(u)
then y = f [g(u)] is called the function of a function or a
composite function.
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8. Types of Functions - Continued
5. periodic Function – A function f(x) in which the range of
the independent variable can separated into equal sub-
intervals such that the graph of the function is the same in
each part then it is called a periodic function.
Symbolically, if f(x+p) = f(x) for all x then p is the period of
f(x).
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9. Illustration 1
If f(x) = x2 – 5, then f is equal to
(a) 0 (b) 5 (c) 10 (d) None of these
Solution:
f(x) = x2 – 5
f = - 5
= 0
9
( )5
( )5 ( )2
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10. Illustration 2
If f(x) = 3 – x2 then f(x) is
(a) An odd function (b) a periodic function
(c) an even function (d) none of these
Solution:
f(x) = 3 – x2
f(-x) = 3 – (-x)2
= 3 - x2
= f(x)
As f(-x) = f(x)
Therefore f(x) = 3 – x2 is an even function.
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11. Illustration 3
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12. Illustration 3 - Continued
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13. Illustration 4
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14. Illustration 4 - Continued
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15. Illustration 5
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16. Illustration 5 - Continued
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17. Illustration 6
If f(x) = logx, (x>0) then f(p) + f(q) + f(r) is
(a) f(pqr) (b) f(p)f(q)f(r) (c) f(1/pqr) (d) None of these
Solution:
If f(x) = log x
then f(p) = log p, f(q) = log q and f(r) = log r
Therefore f(p) + f(q) + f(r) = logp + logq + log r
= log(pqr)
= f(pqr)
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18. Illustration - 7
If f(x) and g(x) are two functions of x such that f(x) + g(x) =
ex and f(x) - g(x) = e-x, then
(a) f(x) is an odd function (b) g(x) is an odd function
(c) f(x) is an even function (d) g(x) is an odd function
Solution:
Now f(x) + g(x) = ex
and f(x) - g(x) = e-x
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19. Illustration – 7 –Continued
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20. Thank you
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