Question

How do you find the domain and range of `(3x+2)/(4x-5)`?

Answer 1

`x inRR,x!=5/4`
`y inRR,y!=3/4`

Explanation:

`"for "y=(3x+2)/(4x-5)`

The denominator cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be.

`"solve "4x-5=0rArrx=5/4larrcolor(red)" excluded value"`

`rArr"domain is "x inRR,x!=5/4`

`"to find any excluded values in the range rearrange making"`
`"x the subject"`

`rArry(4x-5)=3x+2larrcolor(blue)" cross-multiplying"`

`rArr4xy-5y=3x+2`

`rArr4xy-3x=2+5ylarrcolor(blue)" collect terms in x"`

`rArrx(4y-3)=2+5ylarrcolor(blue)" factor out x"`

`rArrx=(2+5y)/(4y-3)`

`"the denominator cannot equal zero"`

`"solve "4y-3=0rArry=3/4larrcolor(red)" excluded value"`

`rArr"range is "y inRR,y!=3/4`