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Compositions of Functions Compositions of Functions Composition functions are used to combined the two functions by using the certain formulas. Let A, B and C be any three sets and let f : A → B and g : B → C be any two functions. Note that the domain of g is the co-domain of f. Define a new function (gof): A → C such that (gof) (a) = g(f(a)) for all a ε A. Here f(a) is an element of B. therefore g(f(a)) is meaningful. The function gof is called the composition of two functions f and g. Note: The small circle o in gof denotes the composition of g and f. Composition of Functions Examples Consider the following examples for the composition of functions Know More About Algebra Questions And Answers


Example 1: Let A = {1, 2}, B = {3, 4} and C = {5, 6} and f : A → B and g : B → C such that f(1) = 3, f(2) = 4, g(3) = 5, g(4) = 6. Find gof. Solution: gof is a function from A → C. Identify the images of elements of A under the function gof. (gof) (1) = g(f(1)) = g(3) = 5 (gof) (2) = g(f(2)) = g(4) = 6 i.e. image of 1 is 5 and image of 2 is 6 under gof .That is, gof = {(1, 5), (2, 6)} Note: For the above definition of f and g, we can't find fog. For some functions f and g, we can find both fog and gof. In certain cases fog and gof are equal. In general fog ≠ gof i.e. the composition of functions need not be commutative always. Example 2: The two functions f, g : R → R are defined by f(x) = x2 + 1, g(x) = x - 1. Find fog and Learn More About Factoring Polynomial


gof and show that fog ≠ gof. Solution: (fog) (x) = f(g(x)) = f(x - 1) = (x - 1)2 + 1 = x2 - 2x + 2 (gof) (x) = g(f(x)) = g(x2 + 1) = (x2 + 1) - 1 = x2 Thus (fog) (x) = x2 - 2x + 2 (gof) (x) = x2 fog ≠ gof Composition of Functions Practice Problems Problem 1: Let f, g: R → R be defined by f(x) = $\frac{(x – 1)}{2}$, and g(x) =2x-1. Show that (fog) ≠ (gof).


Factorization of Polynomials Factorization of Polynomials refers to factoring a polynomial into irreducible polynomials over a given field. It gives out the factors that together form a polynomial function. A polynomial function is of the form xn + xn -1 + xn - 2 + . . . . + k = 0, where k is a constant and n is a power. Polynomials are expressions that are formed by adding or subtracting several variables called monomials. Monomials are variables that are formed with a constant and a variable of some degree. Examples of monomials are 5x3, 6a2. Monomials having different exponents such as 5x3 and 3x4 cannot be added or subtracted but can be multiplied or divided by them. Any polynomial of the form F(a) can also be written as F(a) = Q(a) x D (a) + R (a) using Dividend = Quotient x Divisor + Remainder.


If the polynomial F(a) is divisible by Q(a), then the remainder is zero. Thus, F(a) = Q(a) x D(a). That is, the polynomial F(a) is a product of two other polynomials Q(a) and D(a). For example, 2t + 6t2 = 2t x (1 + 3t). Variables, Exponents, Parenthesis and Operations (+, -, x, /) play an important role in factoring a polynomial. Factorization by dividing the expression by the GCD of the terms of the given expression: GCD of a polynomial is the largest monomial, which is a factor of each term of the polynomial. It involves finding the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of the terms of the expression and then dividing each term by its GCD. Therefore the factors of the given expression are the GCD and the quotient thus obtained. Example 1: Factorize : 2x3 – 6x2 + 4x. Solution: Factors of 2x3 are 1, 2, x, x2, x3,2x, 2x2, 2x3 Factors of 6x2 are 1, 2,3, 6, x, x2, 2x, 2x2 ,3x, 3x2 ,6x, 6x2 Read More About Online Polynomial Factorizer


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Compositions of Functions