What is the domain and range of f(x) = (x-4) /( x+2)??

1 Answer
Alan P.
Oct 7, 2015

Domain: RR - {-2} or (equivalently) RR|(-oo,-2)uu(-2,+oo)

Range: RR- {1}

Explanation:

f(x)=(x-4)/(x+2) is defined for all Real values of x except when (x+2)=0
color(white)("XXX")rarr except for x=-2
So the Domain is all RR except (-2)

To find the Range consider barf(x), the inverse of f(x)

By definition of inverse
color(white)("XXX")f(barf(x))=x
and therefore
color(white)("XXX")(barf(x)-4)/(barf(x)+2)=x

color(white)("XXX")barf(x)-4 = x*barf(x)+2x

color(white)("XXX")barf(x)-xbarf(x)= 2x+4

color(white)("XXX")barf(x)(1-x) = 2x+4

color(white)("XXX")barf(x)=(2x+4)/(1-x)

Which is defined for all values of x!=1

That is the Domain of barf(x) is RR-{1}
and
Since the Domain of a function is the Range of its inverse
color(white)("XXX")the Range of f(x) is RR-{1}

Related questions
See all questions in Domain and Range of a Function
Impact of this question
1949 views around the world
You can reuse this answer
Creative Commons License