Question

Sketch the graph of f(x)=((x^(3)-1)/(x^(2)-1))​

Answer 1

It's easier to sketch if you simplify
`f(x)`. Factorize the numerator and denominator - simple to do with a difference of cubes or squares.

`(x^3 - 1)/(x^2 - 1) = ((x - 1)(x^2 + x + 1))/((x - 1) (x + 1))`

If
`x\\eq1`, we can cancel the factors of
`x-1`. Note that
`f(1)` itself is undefined. For all intents and purposes, aside from the singularity at
`x=1`, the graph of
`f(x)` will look exactly like the graph of

`(x^2 + x + 1)/(x + 1)`

Now, rewrite this as

`(x(x + 1) + 1)/(x + 1)`

Then when
`x\\eq-1` (note that
`f(-1)` is also undefined), we can cancel
`x+1` to reduce this to

`x + \frac1{x+1}`

On its own, the graph of
`x` is a line through the origin. When
`x` is a large number
`\frac1{x+1}` is small. But as
`x` gets closer to -1, the rational term blows up. Effectively, this means the graph of
`f(x)` looks like
`\frac1{x+1}` around
`x=-1`, and far enough away it looks like
`x`.

See the attached plot for a sketch of these details.