Question

If \[f(2x+(\dfrac{y}{8}),2x-(\dfrac{y}{8}))=xy\] , then \[f(x,y)+f(y,x)\]:
1. \[1\]
B. \[0\]
C. \[2\]
D. \[3\]

Answer 1

Hint: First of all assume the values of the given functions equals to \[m\] and \[n\] then find out the value of \[x\] and \[y\].After that find out the values of \[f(x,y)\] and \[f(y,x)\] by putting the values of \[x\] and \[y\] then add \[f(x,y)\] and \[f(y,x)\] to check which option is correct in the given options.

Complete step by step answer:
In mathematics function is defined as a binary relation between the two sets. A function consists of two sets of objects and a correspondence or rule that associates an object in one of the sets with an object in the other set. With the help of a function a relation can be defined between the thing of one type to the other type by some laws hence can say that a function is a relation of a particular type. Functions are also called a subset of relations.
A function consists of two nonempty sets \[X\] and \[Y\] , and a rule \[f\] that associates each element \[x\] in \[X\] with one and only one element \[y\] in \[Y\] . This is symbolized by \[f:\]\[X\to Y\] and reads the function from \[X\] into \[Y\] .
A function is a rule which maps a number to another unique number. In other words, if we start off with an input, and we apply the function, we get an output.
The input to the function is called the independent variable, and is also called the argument of the function. The output of the function is called the dependent variable.
Now according to the question:
Let \[2x+(\dfrac{y}{8})=m\] and \[2x-(\dfrac{y}{8})=n\]
Now we will find out the value of \[x\] and \[y\]
Adding \[2x+(\dfrac{y}{8})=m\] and \[2x-(\dfrac{y}{8})=n\] we will get
\[\Rightarrow \]\[2x+(\dfrac{y}{8})+2x-(\dfrac{y}{8})=m+n\]
Cancelling out \[(\dfrac{y}{8})\] we will get
\[\Rightarrow \]\[4x=m+n\]
Hence, \[x=\dfrac{m+n}{4}\]
For finding the value of \[y\] put the value \[x=\dfrac{m+n}{4}\] either in \[2x+(\dfrac{y}{8})=m\] or in \[2x-(\dfrac{y}{8})=n\]
Here we are putting the value in \[2x+(\dfrac{y}{8})=m\]
\[\Rightarrow 2(\dfrac{m+n}{4})+\dfrac{y}{8}=m\]
\[\Rightarrow \dfrac{2m+2n}{4}+\dfrac{y}{8}=m\]
Taking LCM we will get:
\[\Rightarrow \dfrac{4m+4n+y}{8}=m\]
\[\Rightarrow 4m+4n+y=8m\]
\[\Rightarrow y=4m-4n\]
Hence, \[y=4(m-n)\]
Now according to the question we have to find the value of \[f(x,y)+f(y,x)\] where \[x=\dfrac{m+n}{4}\] and \[y=4(m-n)\]
\[\Rightarrow \]\[f(x,y)=\dfrac{m+n}{4}\times 4(m-n)\]
\[\Rightarrow \]\[f(x,y)=(m+n)(m-n)\]
We know that \[(a+b)(a-b)={{a}^{2}}-{{b}^{2}}\]
So, \[f(x,y)={{m}^{2}}-{{n}^{2}}\]
And \[f(y,x)={{n}^{2}}-{{m}^{2}}\]
\[\Rightarrow f(x,y)+f(y,x)={{m}^{2}}-{{n}^{2}}+{{n}^{2}}-{{m}^{2}}\]
\[\Rightarrow f(x,y)+f(y,x)=0\]

So, the correct answer is “Option 2”.

Note: The word function is derived from a latin word meaning operation and we know that function is a particular type of relation. Therefore, every function is a relation but every relation is not a function. Any closed shape like a circle cannot be a function, but parabolic and exponential curves are functions.