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Isabelle Simmons
9 July, 02:56
F (x) = 2x - 1 what is odd number
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Harper Briggs
9 July, 05:49
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Either the question is wrong or it is must be "for all positive integers number". Because for real number there are examples where it is not true. For example x = 1/2 it gives 0 which is a even number. For all positive integers the statement is true by following proof:
f (x) = 2x-1
f (x) %2 = (2x-1) %2 = (2x-1+2) %2 = (2x+1) %2 = ((2 * (x%2)) %2 + 1) %2
now x%2 is either 1 or 0
for x%2 = 1
f (x) %2 = ((2 * (1)) %2 + 1) %2 = (2%2+1) %2 = 1
for x%2 = 0
f (x) %2 = ((2 * (0)) %2 + 1) %2 = (0+1) %2 = 1
So in all cases f (x) %2 = 1 so f (x) must be odd for all positive integers
Here is another proof by mathematical induction:
let x = 1 be base condition then
f (1) = 1 so it is true for that
Lets assume f (x) is odd
then f (x+1) = 2 (x+1) - 1
f (x+1) = 2x+2-1
f (x+1) = 2x+1
f (x+1) = 2x-1 + 2
f (x+1) = f (x) + 2
f (x) = odd
so f (x) + 2 must be odd.
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