Question

If a function satisfying f(x) + f(x-1/x) = 1+x, x belongs to R -{0}.
Find f(x).
plz ans it correctly or else your ans will be reported. ​

Answer 1

Answer:

f(x)+f((x-1)/x)=x+1    -------->1

replace x by x-1/x

f((x-1)/x)+f(-1/(x-1))=((x-1)/x)+1=1-(1/x)+1=(x-1/x)+1      --------->2

equation 1 - equation 2

f(x)-f(-1/(x-1))=x-((x-1)/x) ⇒ x-1+(1/x)   -------------3

replace x by(-1/(x-1))

f(-1/(x-1))-f(-1/(-1/(x-1))-1)=(-1/(x-1))-1-((x-1)/1)

f(-1/(x-1))-f((x-1)/x)=(-1/(x-1))-x    ------------4

Adding 1 3 & 4

2f(x)=x+1+x-((x-1)/x)-x-(1/(x-1))

2f(x)=x+1-((x-1)/x)-(1/(x-1))

       =(x(x-1)(x+1)-(x-1)²-x)/(x(x-1))

2f(x)=(x(x²-1)-(x-1)²-x)/(x(x-1))

f(x)=(x³-x-x²-1+2x-x)/(2x(x-1))

f(x)=(x³-x²-1)/(2x(x-1))

Explanation: