Question

How do you find the domain of `f(x)=(x-3)/(x^2-x-2)`?

Answer 1

`x in RR` \ `{-1; 2}`

Explanation:

To find the domain of a rational function, you need to determine all values of `x` for which the demoninator is equal to `0`.

So, set `x^2 - x - 2 = 0` and solve the quadratic equation.

There are several possibilities to solve a quadratic equation, one of my favourite is "completing the circle":

`x^2 - x - 2 = 0`
`<=> x^2 - x = 2`

Try to write the left side of the equation like `(x-a)^2 = x^2 - 2ax + a^2`.
We already have the `x^2`, and from `-x = -2ax` we can conclude that `a = 1/2`. So, the only term that is missing on the left side to complete the quadratic form is `+1/4`.

In order to prevail the equality, the term needs to be added on both sides of the equation, leading us to the following:
`x^2 - x + 1/4 = 2 + 1/4`
`<=> (x-1/2)^2 = 9/4`

Now, we can calculate the root on both sides. Don't forget that the root of `9/4` has two solutions since both `(3/2)^2 = 9/4` and `(-3/2)^2 = 94` hold.

Thus, we can further solve the quadratic equation as follows:
`x - 1/2 = 3/2 or x - 1/2 = - 3/2`
`<=> x = 2 or x = -1`

This means that for those two values, the denominator of our function would be `0` which is not admissible.

So, the domain of the function are all real numbers except `x = 2` and `x = -1`.
`x in RR` \ `{-1; 2}`