Let $$f\left( x \right) = a{x^2} + bx + c,$$ where $$a,\,b,\,c$$ are rational, and $$f:Z \to Z,$$ where $$Z$$ is the set of integers. Then $$a+b$$ is :
A.
a negative integer
B.
an integer
C.
non-integral rational number
D.
none of these
Answer :
an integer
Solution :
$$f:Z \to Z$$ is defined as $$f\left( x \right) = a{x^2} + bx + c$$
which implies for integer inputs, the function gives integer outputs.
$$ \Rightarrow f\left( 0 \right) = c = {Z_1}......\left( 1 \right)$$ (where $${Z_1}$$ is some integer)
Similarly, $$f\left( 1 \right) = a + b + c = {Z_2}......\left( 2 \right)$$ (where $${Z_2}$$ is some integer)
Equation $$\left( 2 \right) - \left( 1 \right)$$ gives $$a + b = {Z_2} - {Z_1},$$ which is also an integer.
Releted MCQ Question on Calculus >> Function
Releted Question 1
Let $$R$$ be the set of real numbers. If $$f:R \to R$$ is a function defined by $$f\left( x \right) = {x^2},$$ then $$f$$ is: