Question

How do you find the domain and range of `f(x)=(2x-1)/(3-x)`?

Answer 1

Perform polynomial division on `f(x)` to put it into partial fraction form. From this, you can determine asymptotes which help you to determine domain and range.

Explanation:

For a rational function of the form `f(x) = (g(x)) / (h(x))` such as the one above, `f(x)=(2x-1)/ (3-x)`, since the degree of the polynomial is the same in the denominator as it is in the numerator, you must divide through. Doing so, we get `f(x) = -2 + (5) / (3 -x)`.

From this it is evident that this is a rectangular hyperbola with asymptotes at `x = 3` and `y = -2`, so neither of these are included in the domain or range respectively.

Therefore we get, `dom f in (-oo, 3) uu (3, oo)` and `ran f in (-oo, -2) uu (-2, oo)`.

Answer 2

`x inRR,x!=3`
`y inRR,y!=-2`

Explanation:

`"f(x) is defined for all real values of x apart from values that "`
`"make the denominator zero"`

`"Equating the denominator to zero and solving gives the value"`
`"that x cannot be"`

`"solve " 3-x=0rArrx=3larrcolor(red)" excluded value"`

`rArr"domain is "x inRR,x!=3`

`"to find any excluded values in the range rearrange y = f(x)"`
`"making x the subject"`

`rArry(3-x)=2x-1`

`rArr3y-xy=2x-1`

`rArr-xy-2x=-(1+3y)`

`rArrx(-y-2)=-(1+3y)`

`rArrx=-(1+3y)/(-y-2)`

`"the denominator cannot equal zero"`

`"solve " -y-2=0rArry=-2larrcolor(red)"excluded value"`

`rArr"range is " y inRR,y!=-2`