F (x) = 2x - 1 what is odd number

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.