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This is the Exam of PreCalculus which includes Irrational Number, Decimal Expansion, Rational, Equations, Polynomials, Prime, Factored, Integer Coe±Cients, Explicit Factorisation etc. Key important points are: Integer Function, Function, Greatest, Two Functions, Domains, Identity Function, One to One, Domain, Complete Factorization, Zeros
Typology: Exams
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Question Score Max 1 9 2 8 3 7 4 5 5 6 6 4 7 10 8 3 9 4 Total 57
(1) [Marks: 9] Sketch the graph of the function f (x). Remember that bxc denotes the greatest integer function.
(3) [Marks: 7] Find two functions f, g, neither of which is the identity function I(x) = x, such that F = f ◦ g where F (x) = x + 3 − √x + 3 + 1^2 + 1
(4) [Marks: 5] (a) Prove that f (x) = (^) xx^ + 1− 1 is one-to-one.
(b) Prove that g(x) = x x 22 + 1− 1 is not one-to-one.
(5) [Marks: 6] Find f −^1 (x) for the function f (x) = x^2 − 4 x + 3 with domain (−∞, 2]
(6) [Marks:4] 5 is a zero of f (x) = 4x^3 − 20 x^2 − x + 5. Find all the other zeros and give the complete factorization of f (x).
(8) [Marks: 3] Find all the roots (including complex roots) of p(x) = x^3 + 2x^2 + 3x.
(9) [Marks: 4] Find the partial fraction decomposition of g(x) = (^) x(x − 2)(2^1 x − 1).