f(X)= (x^2)/(x^2 - 1) .find the symmetry , intercepts, asymptotes, interval of increase and decrease, local extreme values, concavity, point of deflection and draw the graph of the function.kindly show the working.

Answer 1

review your basic rules in this stuff.

f = x^2/(x^2-1)
f' = -2x/(x^2-1)^2
f" = 2(3x^2+1)/(x^2-1)^3

I expect you can find intercepts ok.
Since f(x) is even, it's symmetric about the y-axis.

vertical asymptotes where x^2-1 = 0

Since f = 1 + 1/(2(x^2-1))
horizontal asymptote is at y=1

f is increasing where f' > 0
f is concave up where f" > 0

You can check your calculations by viewing the graph at

http://www.wolframalpha.com/input/?i=+x^2%2F%28x^2-1%29