How do you graph #g(x) = -9/(x-9)+6#? What is the domain and range?

2 Answers
Abdelhalim Mira
Apr 22, 2017

graph{(6x-63)/(x-9) [-75.7, 84.3, -37.2, 42.8]}

domain : #R-{9}#
range : #R-{6}#

Explanation:

simplify the function : #g(x)=6-(9/(x-9))=((6(x-9))/(x-9))-9/(x-9)#

#=(6x-54-9)/(x-9)=(6x-63)/(x-9)#

the function has horizontal asymptote at #y=6#

and vertical asymptote at #x=9#

so the #x# never reaches 9, #y# never reaches 6

Jim G.
Apr 22, 2017

see explanation.

Explanation:

The denominator of g(x) cannot be zero as this would make g(x) #color(blue)"undefined".# Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

#"solve " x-9=0rArrx=9" is the asymptote"#

#rArr" domain is " x inRR,x!=9#

Horizontal asymptotes occur as

#lim_(xto+-oo),g(x)toc" ( a constant)"#

divide terms on numerator/denominator by x

#g(x)=-(9/x)/(x/x-9/x)+6=-(9/x)/(1-9/x)+6#

as #xto+-oo,g(x)to-0/(1-0)+6#

#rArry=6" is the asymptote"#

#rArr"range is " y inRR,y!=6#

#color(blue)"Intercepts"#

#x=0toy=-9/(-9)+6=7larrcolor(red)" y-intercept"#

#y=0to-9/(x-9)+6=0#

#rArr9/(x-9)=6rArrx=21/2larrcolor(red)" x-intercept"#
graph{-(9/(x-9))+6 [-25.66, 25.65, -12.83, 12.83]}

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