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"Scientific Approach"
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maths - Relations & Functions Advanced Formulas RELATION RECALL २. DOMAIN, CODOMAIN & RANGE 1 A relation R from set A to set B is Domain Set of all first elements > subset of X B. A Number of relations from A to B = 2nm Codomain Set of all second elements ళ (where n = elements in A m = elements in B) Range g Codomain 3. TYPES OF RELATIONS ४. FUNCTIONS ADVANCED TYPES () Reflexive Relation: (,0) C R (a) Even Function: f(-x) = f(x) for qll qA (b) Odd Function: f(-x) = -f(x) (b) Symmetric Relqtion: 9*>3 If (0,b) CR = (b,0) CR (c) Periodic Function: 1(x+1) (c) Transitive Relation: If (a b) & R f(x+T) = f(x) and (b c)eR = (d c)e R (d) Monotonic Functions: Increasing; (d) Equivdlence Relqtion: Decreasing Reflexive + Symmetric + Tronsitive A FUNCTION 5. COMPOSITION OF FUNCTIONS 6. INVERSE OF f has an inverse if and A function 9(f (g only if f is bijective ५=X Condition: Range of f = Domdin of g (fo f-1)(X) = X (f-1 . f)(x) = X ALGEBRA OF FUNCTIONS 7. (f+ g)(x) = f(x) + g(x) DOMAIN RULES (IMPORTANT) 8 (f-g)(x) = f(x) -g(x) Denominator # 0 (fg)(x) f(x)g(x) Even root > 0 (g)(x) = g@) g(x) # 0 Logarithm argument > 0 10. IMPORTANT RESULTS 9. GRAPH TRANSFORMATIONS upward shift Inverse of inverse original function f(x)+k -> Composition is ossociotive but not f(X - q) _> right shift commutative -f(x) -> reflection in Xzaxis Every inverse is unique f(-x) -> reflection in y-axis Relations & Functions Advanced Formulas RELATION RECALL २. DOMAIN, CODOMAIN & RANGE 1 A relation R from set A to set B is Domain Set of all first elements > subset of X B. A Number of relations from A to B = 2nm Codomain Set of all second elements ళ (where n = elements in A m = elements in B) Range g Codomain 3. TYPES OF RELATIONS ४. FUNCTIONS ADVANCED TYPES () Reflexive Relation: (,0) C R (a) Even Function: f(-x) = f(x) for qll qA (b) Odd Function: f(-x) = -f(x) (b) Symmetric Relqtion: 9*>3 If (0,b) CR = (b,0) CR (c) Periodic Function: 1(x+1) (c) Transitive Relation: If (a b) & R f(x+T) = f(x) and (b c)eR = (d c)e R (d) Monotonic Functions: Increasing; (d) Equivdlence Relqtion: Decreasing Reflexive + Symmetric + Tronsitive A FUNCTION 5. COMPOSITION OF FUNCTIONS 6. INVERSE OF f has an inverse if and A function 9(f (g only if f is bijective ५=X Condition: Range of f = Domdin of g (fo f-1)(X) = X (f-1 . f)(x) = X ALGEBRA OF FUNCTIONS 7. (f+ g)(x) = f(x) + g(x) DOMAIN RULES (IMPORTANT) 8 (f-g)(x) = f(x) -g(x) Denominator # 0 (fg)(x) f(x)g(x) Even root > 0 (g)(x) = g@) g(x) # 0 Logarithm argument > 0 10. IMPORTANT RESULTS 9. GRAPH TRANSFORMATIONS upward shift Inverse of inverse original function f(x)+k -> Composition is ossociotive but not f(X - q) _> right shift commutative -f(x) -> reflection in Xzaxis Every inverse is unique f(-x) -> reflection in y-axis - ShareChat