Question 6, Let f P3 be defined by the rule f(x) 2x3 + 4x2 + x. Consider the fol

Question

Question 6, Let f P3 be defined by the rule f(x) 2x3 + 4x2 + x. Consider the following statement: Va EZ, (if f(a) is odd, then a is odd) Prove this statement in three ways: 1. Use proof by contradiction (that is to say, assume that the negation of the original statement is true and find a contradiction. You will have to negate the quantified statement.) 2. Use proof by contrapositive (here, we mean only apply the contrapositive argument to the "if-then" appearing inside of the parentheses, leaving the "VaEZ" as it is.) 3. Give a direct proof. (Hint: Solve for a.)

Explanation / Answer

f(x) = 2x^3 + 4x^2 + x = 2(x^3 + 2x^2) + x

we assume if f(a) is odd then a is not odd

a = 2k

f(a) = 2 (a^3 + 2a^2) + a = 2 m + 2 k

where m = (a^3 + 2a^2)  

= 2 (m+k)

here f(a) is even , which contradicts that f(a) is odd

hence Proved

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