Question

What is the range of the function `f(x)=(5x-3)/(2x+1)`?

Answer 1

The range is `y in RR-{5/2}`

Explanation:

`f(x)=(5x-3)/(2x+1)`
Let

`y=(5x-3)/(2x+1)`

`y(2x+1)=5x-3`

`2yx+y=5x-3`

`5x-2yx=y+3`

`x(5-2y)=(y+3)`

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

The domain of `x=f(y)` is `y in RR-{5/2}`

This is also `f^-1(x)=(x+3)/(5-2x)`

graph{(5x-3)/(2x+1) [-22.8, 22.83, -11.4, 11.4]}

Answer 2

`y inRR,y!=5/2`

Explanation:

`"given "y=(5x-3)/(2x+1)`

`"rearrange making x the subject"`

`rArry(2x+1)=5x-3larrcolor(blue)"cross-multiplying"`

`rArr2xy+y=5x-3larrcolor(blue)" distributing"`

`rArr2xy-5x=-3-ylarrcolor(blue)" collect terms in x"`

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

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

`"the denominator cannot equal zero as this would"`
`"be undefined"`

`2y-5=0rArry=5/2larrcolor(red)" excluded value"`

`"range is "y inRR,y!=5/2`