Limits and Continuity – Intuitive Approach Chapter 8 Paper 4: Quantitative Aptitude- Mathematices

Ms. Ritu Gupta B.A. (Hons.) Maths and MA (Maths)

Introduction to Function

• Fundamental Knowledge • Its application

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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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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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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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Types of Functions - Continued

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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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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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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

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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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Illustration 3

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Illustration 3 - Continued

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Illustration 4

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Illustration 4 - Continued

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Illustration 5

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Illustration 5 - Continued

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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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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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Illustration – 7 –Continued

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Thank you

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