Question

If we have two sets as A =\left\\{ {x:f(x) = 0} \right\\} and B =\left\\{ {x:g(x) = 0} \right\\} then A$\cap$ B will be the root of
A. $\dfrac{{f(x)}}{{g(x)}} = 0$
B. $\dfrac{{g(x)}}{{f(x)}} = 0$
C. ${\left| {f(x)} \right|^2} + {\left| {g(x)} \right|^2} = 0$
D. None of these.

Answer 1

In this question, we will assume a function and find the values in set which satisfies f(x) and then find the elements in set B which satisfy g(x). After this, we will find the elements common to set A and B. Finally we check which option satisfies for the given roots.

Complete step-by-step answer:
Let us assume the function as $f(x) = (x – a)(x – b)$
And function $g(x) = (x –b)(x – c)$.
We know that if $f(p) = 0$ then ‘p’ is the root of the equation $f(x)=0$.
Now, $f(a) = 0$ and $f(b) =0$. So, we can say that ‘a’ and ‘b’ are roots of the equation f(x)=0.
Similarly, $g(b) = 0$ and $g(c) =0$. So, we can say that ‘b’ and ‘c’ are roots of the equation $g(x)=0$.
Therefore, we can write:
A=\left\\{ {a,b} \right\\} and B =\left\\{ {b,c} \right\\}.
Now we will calculate the elements common to A and B both.
A$\cap$ B = \left\\{ b \right\\}.
Now we will check the option to find the function whose root is the A$\cap$ B.
Let us take option A:
$\dfrac{{f(x)}}{{g(x)}}$=$\dfrac{{(x - a)(x - b)}}{{(x - b)(x - c)}}$ . Putting the value of x= b, we get:
$\dfrac{{f(x)}}{{g(x)}}$=$\dfrac{0}{0}$ which is an indeterminate form. So, A$\cap$ B is not the root of $\dfrac{{f(x)}}{{g(x)}}$=0.
Similarly, we can say that x = A$\cap$ B is not the root of $\dfrac{{g(x)}}{{f(x)}}$ as it gives undefined value.
Let us take option C:
${\left| {f(x)} \right|^2} + {\left| {g(x)} \right|^2}$=${(x - a)^2}{(x - b)^2} + {(x - b)^2}{(x - c)^2}$=0 . Putting the value of x= b, we get:
LHS= ${(b - a)^2}{(b - b)^2} + {(b - b)^2}{(b - c)^2} = 0$. It means that x= A$\cap$ B is the root of the equation ${\left| {f(x)} \right|^2} + {\left| {g(x)} \right|^2} = 0$
So, option C is the correct option.

Note: The important thing in the question is that you should first understand the meaning of each of the notation like A =\left\\{ {x:f(x) = 0} \right\\} means that A is the set of values of ‘x’ which satisfies f(x) = 0. Similarly we can also find an equation whose root is A$\cup$ B. x= A$\cup$ B is the root of equation f(x)g(x) =0.