Let a, b, c be rational numbers and f: Z → Z be a function given by $f(x) = ax^2 + bx + c$. Then, $a + b$ is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → an integer
Since f: Z → Z is given by
$f(x) = ax^2 + bx + c$ for all $x ∈ Z$.
$∴ f(x) = ax^2 + bx + c$ takes integral values for all $x ∈ Z$
⇒ f(x) is an integer for all $x ∈ Z$
⇒ f(0) and f(1) are integers.
$⇒ f(1) - f(0)$ is an integer.
$⇒ (a+b+c) -c$ is an integer [$∵ f(0) = c$ and $f(1) = a+b+c$]
⇒ a+b is an integer.