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Mathematics
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Session Functions, Limits and Continuity-1
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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
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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.
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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.
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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, 16... }
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Example– 1 Find the domain of the following functions:
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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)
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Example- 2 Find the range of the following functions:
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-1 cos2x 1 for all xR -3 3cos2x 3 for all xR -2 1 + 3cos2x 4 for all xR -2 f(x) 4 Hence, range of f = [-2, 4] Example – 2(ii)
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Some Standard Real Functions (Constant Function) O Y X (0, c) f(x) = c Domain = R Range = {c}
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Domain = R Range = R Identity Function X Y O 45 0 I(x) = x
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Modulus Function f(x) = x f(x) = - x O X Y
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y = sinx y = |sinx| Example
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Greatest Integer Function = greatest integer less than or equal to x.
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Algebra of Real Functions Multiplication by a scalar: For any real number k, the function kf is defined by
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Algebra of Real Functions (Cont.)
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Composition of Two Functions
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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
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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
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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
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Example – 5 Prove that is an even function.
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Example - 6 Let the function f be f(x) = x 3 - kx 2 + 2x, xR, 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 xR 2kx 2 = 0 for all xR k = 0 -x 3 - kx 2 - 2x = - x 3 + kx 2 - 2x for all xR
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Limit of a Function x2.52.62.72.82.92.993.013.13.23.33.43.5 f(x)5.55.65.75.85.95.996.016.16.26.36.46.5 As x approaches 3 from left hand side of the number line, f(x) increases and becomes close to 6
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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
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x takes the values 2.91 2.95 2.9991.. 2.9999 ……. 9221 etc. Left Hand Limit x 3 Y O X
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x takes the values 3.1 3.002 3.000005 …….. 3.00000000000257 etc. Right Hand Limit 3 X Y O x
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Existence Theorem on Limits
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Example – 7 Which of the following limits exist:
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Example - 7 (ii)
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Properties of Limits If and where ‘m’ and ‘n’ are real and finite then
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The limit can be found directly by substituting the value of x. Algebraic Limits (Substitution Method)
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Algebraic Limits (Factorization Method) When we substitute the value of x in the rational expression it takes the form
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Algebraic Limits (Rationalization Method) When we substitute the value of x in the rational expression it takes the form
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Standard Result If n is any rational number, then
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Example – 8 (i)
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Example – 8 (ii)
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Example – 8 (iii)
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Solution Cont.
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Example – 8 (iv)
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Example – 8 (v)
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Thank you
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