Question

What is the domain and range of `y=(-4x-3)/(x-2)`?

Answer 1

Domain:

The domain of any rational function will be influenced by vertical asymptotes. Vertical asymptotes are found by setting the denominator to zero then solving:

`x - 2 = 0`

`x = 2`

Hence, there will be a vertical asymptote at `x = 2`. Therefore, the domain will be `{x|x !=2, x ="all real numbers"}`.

Range:

The range of any rational function will be influenced by the existence of horizontal asymptotes. Since the degree of the denominator is equal to that of the numerator, the asymptote occurs at the ratio between the coefficients of the terms of highest degree.

`(-4x)/x -> -4/1 ->-4`

Hence, there will be a horizontal asymptote at `y = -4`.

The range is therefore `{y|y!=-4, y ="all real numbers"}`.

Hopefully this helps!