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

##### 2 Answers

Perform polynomial division on

#### Explanation:

For a rational function of the form

From this it is evident that this is a rectangular hyperbola with asymptotes at

Therefore we get,

#### 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#