Question

What is the domain and range of `f(x) =( x^2 - x - 6) / (x^2 + x - 12)`?

Answer 1

Domain is all values except `x=-4` and `x=3` range is from `1/2` to `1`.

Explanation:

In a rational algebraic function `y=f(x)`, domain means all values that `x` can take. It is observed that in the given function `f(y)=(x^2-x-6)/(x^2+x-12)`, `x` cannot take values where `x^2+x-12=0`

Factorizing this becomes `(x+4)(x-3)=0`. Hence domain is all values except `x=-4` and `x=3`.

Range is values that `y` can take. Although, one may have to draw a graph for this, but here as `x^2-x-6=(x-3)(x+2)` and hence

`f(y)=(x^2-x-6)/(x^2+x-12)=((x-3)(x+2))/((x+4)(x-3))=(x+2)/(x+4)`

= `1-2/(x+4)`

and hence range is from `1/2` to `1`.