Given if f(x)=x^2-4 and g(x)=sqrt(x-3) how do you find f(g(x)) and g(f(x))?

2 Answers
Dec 5, 2016

The answers are f(g(x))=x-7
and g(f(x))=sqrt(x^2-7)

Explanation:

This is a composition of functions.

f(x)=x^2-4

g(x)=sqrt(x-3)

fog(x)=f(g(x))=f(sqrt(x-3))=(sqrt(x-3))^2-4

=x-3-4=x-7

And

gof(x)=g(f(x))=g(x^2-4)=sqrt((x^2-4)-3)

=sqrt(x^2-4-3)=sqrt(x^2-7)

You can see that

f(g(x))!=g(f(x))

Dec 5, 2016

f(g(x))=x-7

g(f(x))=sqrt(x^2-7)

Explanation:

Put g inside f to find f(g(x))

f(x)=color(red)x^2-4

g(x)=color(red)sqrt(x-3)

f(g(x))=(color(red)sqrt(x-3))^2-4

color(white)(f(g(x)))=x-3-4

color(white)(f(g(x)))=x-7

Put f inside g to find g(f(x))

g(x)=sqrt(color(red)x-3)

f(x)=color(red)(x^2-4)

g(f(x))=sqrt(color(red)(x^2-4)-3)

color(white)(g(f(x)))=sqrt(x^2-7)