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

2 Answers
May 30, 2017

Answer:

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)#.

May 30, 2017

Answer:

#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#