How do you find the compositions given #f(x)= sqrt(x-2)# and #g(x)= x^2-1#?

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Answer 1 · 2016-01-11T12:33:03

#f@g=f(g(x))=sqrt(x^2-3)#;
#g@f=g(f(x))=x-3;#

where:

#y=g(x)#

then we can think:

#z=f(g(x))=f(y)=sqrt(y-2)#

with
#y=g(x)=x^2-1#

#:.f@g=f(g(x))=sqrt((x^2-1)-2)=sqrt(x^2-1-2)=#
#=sqrt(x^2-3)#

where:

#y=f(x)=sqrt(x-2)#

then we can think:

#z=g(f(x))=g(y)=y^2-1#

#:.g@f=g(f(x))=(sqrt(x-2))^2-1=x-2-1=x-3#