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

1 Answer
Jan 11, 2016

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

Explanation:

The Function Composition tells us that:

  1. z=f@g=f(y)=f(g(x))

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)

  1. z=g@f=g(f(x))

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