Find the range of `f(x)=x^2-x-3.`

Explanation

`f(x)=x^(2)-x-3`
`=(x-(1)/(2))^(2)-(1)/(4)-3=(x-(1)/(2))^(2)-(13)/(4)`
Now `(x-(1)/(2))^(2) ge 0 AA x in R`
`implies(x-(1)/(2))^(2)-(13)/(4) ge (-13)/(4) AA x in R`
Hence range is `[-(13)/(4), oo)`
Alternatively, let `y=x^(2)-x-3`
` :. x^(2)-x-3-y=0`
Since x is real, `D ge 0`
` :. (-1)^(2)-4(-3-y) ge 0`
`implies 13+4y ge 0`
`implies y ge -13//4`
Therefore, range is `[-(13)/(4),oo)`