What is the domain and range of #f(x)=(x^2-9)/(x^2-25)#?

1 Answer
Jim G.
Jun 28, 2017

#x inRR,x!=+-5#
#y inRR,y!=1#

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be.

#"solve " x^2-25=0rArr(x-5)(x+5)=0#

#rArrx=+-5larrcolor(red)" are excluded values"#

#rArr"domain is " x inRR,x!=+-5#

#"to find any excluded value in the range we can use the"#
#"horizontal asymptote"#

#"horizontal asymptotes occur as"#

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

divide terms on numerator/denominator by the highest power of x, that is #x^2#

#f(x)=(x^2/x^2-9/x^2)/(x^2/x^2-25/x^2)=(1-9/x^2)/(1-25/x^2)#

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

#rArry=1" is the asymptote and thus excluded value"#

#rArr"range is " y inRR,y!=1#

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