Mathematics

Session Functions, Limits and Continuity-1

 Function  Domain and Range  Some Standard Real Functions  Algebra of Real Functions  Even and Odd Functions  Limit of a Function; Left Hand and Right Hand Limit  Algebraic Limits : Substitution Method, Factorisation Method, Rationalization Method  Standard Result Session Objectives

Function If f is a function from a set A to a set B, we represent it by If A and B are two non-empty sets, then a rule which associates each element of A with a unique element of B is called a function from a set A to a set B. If f associates then we say that y is the image of the element x under the function or mapping and we write Real Functions: Functions whose co-domain, is a subset of R are called real functions.

Domain and Range The set of the images of all the elements under the mapping or function f is called the range of the function f and represented by f(A). The set A is called the domain of the function and the set B is called co-domain.

Domain and Range (Cont.) For example: Consider a function f from the set of natural numbers N to the set of natural numbers N i.e. f : N N given by f(x) = x 2 Domain is the set N itself as the function is defined for all values of N. Range is the set of squares of all natural numbers. Range = {1, 4, 9, }

Example– 1 Find the domain of the following functions:

The function f(x) is not defined for the values of x for which the denominator becomes zero Hence, domain of f = R – {1, 2} Example– 1 (ii)

Example- 2 Find the range of the following functions:

-1  cos2x  1 for all xR -3  3cos2x  3 for all xR -2  1 + 3cos2x  4 for all xR  -2  f(x)  4 Hence, range of f = [-2, 4] Example – 2(ii)

Some Standard Real Functions (Constant Function) O Y X (0, c) f(x) = c Domain = R Range = {c}

Domain = R Range = R Identity Function X Y O 45 0 I(x) = x

Modulus Function f(x) = x f(x) = - x O X Y

y = sinx y = |sinx| Example

Greatest Integer Function = greatest integer less than or equal to x.

Algebra of Real Functions Multiplication by a scalar: For any real number k, the function kf is defined by

Algebra of Real Functions (Cont.)

Composition of Two Functions

Let f : R  R + such that f(x) = e x and g(x) : R +  R such that g(x) = log x, then find (i) (f+g)(1) (ii) (fg)(1) (iii) (3f)(1) (iv) (fog)(1) (v) (gof)(1) (i) (f+g)(1) (ii) (fg)(1) (iii) (3f)(1) = f(1) + g(1) =f(1)g(1) =3 f(1) = e 1 + log(1) =e 1 log(1) =3 e 1 = e + 0 = e x 0 =3 e = e = 0 Example - 3 Solution : (iv) (fog)(1) (v) (gof)(1) = f(g(1)) = g(f(1)) = f(log1) = g(e 1 ) = f(0) = g(e) = e 0 = log(e) =1 = 1

Find fog and gof if f : R  R such that f(x) = [x] and g : R  [-1, 1] such that g(x) = sinx. Solution: We have f(x)= [x] and g(x) = sinx fog(x) = f(g(x)) = f(sinx) = [sin x] gof(x) = g(f(x)) = g ([x]) = sin [x] Example – 4

Even and Odd Functions Even Function : If f(-x) = f(x) for all x, then f(x) is called an even function. Example: f(x)= cosx Odd Function : If f(-x)= - f(x) for all x, then f(x) is called an odd function. Example: f(x)= sinx

Example – 5 Prove that is an even function.

Example - 6 Let the function f be f(x) = x 3 - kx 2 + 2x, xR, then find k such that f is an odd function. Solution: The function f would be an odd function if f(-x) = - f(x)  (- x) 3 - k(- x) 2 + 2(- x) = - (x 3 - kx 2 + 2x) for all xR  2kx 2 = 0 for all xR  k = 0  -x 3 - kx 2 - 2x = - x 3 + kx 2 - 2x for all xR

Limit of a Function x f(x) As x approaches 3 from left hand side of the number line, f(x) increases and becomes close to 6

Limit of a Function (Cont.) Similarly, as x approaches 3 from right hand side of the number line, f(x) decreases and becomes close to 6

x takes the values …… etc. Left Hand Limit x 3 Y O X

x takes the values …… etc. Right Hand Limit 3 X Y O x

Existence Theorem on Limits

Example – 7 Which of the following limits exist:

Example - 7 (ii)

Properties of Limits If and where ‘m’ and ‘n’ are real and finite then

The limit can be found directly by substituting the value of x. Algebraic Limits (Substitution Method)

Algebraic Limits (Factorization Method) When we substitute the value of x in the rational expression it takes the form

Algebraic Limits (Rationalization Method) When we substitute the value of x in the rational expression it takes the form

Standard Result If n is any rational number, then

Example – 8 (i)

Example – 8 (ii)

Example – 8 (iii)

Solution Cont.

Example – 8 (iv)

Example – 8 (v)

Thank you