Can one show me how to plot this graph by hand (composition).

Question

Can one show me how to plot this graph by hand (composition).

$$\frac{1}{1-x^2}$$

Plot[1/(1 - x^2), {x, -5, 5}]

Output result see picture in the end

Fine, thanks, I got it by plot 1/(1 + x) and 1/(1 - x)

Plot[{1/(1 - x), 1/(1 + x), 1/(1 - x) 1/(1 + x)}, {x, -5, 5}]

Output result see picture in the end

Answer 1

$$\frac{1}{1-x^2}$$ has the domain $\Bbb{R}-\{-1,1\}$

If $x\to\{-1^{+},1^{-}\}$, then $\frac{1}{1-x^2}$ approaches $+\infty$, and if $x\to\{-1^{-},1^{+}\}$, then $\frac{1}{1-x^2}$ approaches $-\infty$

For $x\in(-\infty,-1)\cup (1,\infty)$, $\frac{1}{1-x^2}$ is negative, and tends to $0$ as $x\to\pm\infty$

This should give you some hints as to the general shape of the graph.

For greater accuracy, you may calculate the derivative of the graph at various points.

Answer 2

The tag does not mention calculus, so we avoid mentioning the derivative,

First note the symmetry: If $f(x)=\frac{1}{1-x^2}$ then $f(-x)=f(x)$. So there is mirror symmetry across the $y$ axis. This means we need only plot our function for $x\ge 0$. For the rest, we reflect across the $y$ axis. Who can resist a two for one deal?

From now on we restrict attention to $x\ge 0$. We have $f(0)=0$. That's the only point we will explicitly plot.

Now suppose that $x$ increases towards $1$. Then $1-x^2$ decreases towards $0$. So $\frac{1}{1-x^2}$ increases. When $x$ is near $1$ but slightly less than $1$, the number $\frac{1}{1-x^2}$ is very large negative. That means that the line $x=1$ is an asymptote to the curve $y=\frac{1}{1-x^2}$.

Our function is not defined at $x=1$. That sort of information is often very useful.

When $x\gt 1$ but $x$ is very near $1$, the number $1-x^2$ is negative, but very close to $0$. Thus $\frac{1}{1-x^2}$ is huge negative. Now let $x$ increase. Then fairly quickly $1-x^2$ becomes large negative, so $\frac{1}{1-x^2}$ is negative but close to $0$. Thus the positive $x$-axis is an asymptote to our curve. For $x$ quite a bit greater than $1$, the curve is below the positive $x$-axis, but after a while visually indistinguishable from it.

This information should be enough to draw a pretty good curve. To get an idea of the scale, it may be useful to find $f(x)$ at say $x=1/2$, $x=2$, and $x=10$.