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# Graph sketching (Tomato Subjective 124)

Problem: Sketch on plain paper, the graph of $$y = \frac {x^2 + 1} {x^2 – 1}$$

Discussion:

There are several steps to find graph of a function. We will use calculus to analyze the function. Here y=f(x)

1. Domain: The function is defined at all real numbers except x =1 and x = -1 which makes the denominator 0.
2. Even/Odd: Clearly f(x) = f(-x). Hence it is sufficient to investigate the function for positive values of x and then reflect it about y axis.
3. Critical Points: Next we investigate the critical points. Critical Points are those values of x for which the first derivative of f(x) is either 0 or undefined. Since $$\displaystyle{y = \frac {x^2 + 1} {x^2 – 1}}$$, then $$\displaystyle{f'(x) = \frac {\left(\frac{d}{dx}(x^2 + 1)\right)(x^2 -1) – \left(\frac{d}{dx}(x^2 – 1)\right)(x^2 + 1)} {(x^2 – 1)^2} }$$.
This implies $$\displaystyle{f'(x) = \frac{2x^3 – 2x – 2x^3 – 2x}{(x^2 – 1)^2 } = -\frac{4x}{(x^2 – 1)^2 }}$$
Hence critical points are x =0 , 1, -1
4. Monotonicity: The first derivative is negative for all positive values of x (note that we are only investigating for positive x values, since we can then reflect the picture about y axis as previously found). Hence the function is ‘decreasing’ for all positive value of x.