Composition of inverse functions f(x) = x^2+5, x=5 for f(g(x)) I get x but for g(f(x)) I get -x I guess this makes sense looking at the domain and range of f(x), but it bothers me that I'm getting two different values(x and -x).

Question

Composition of inverse functions

f(x) = x^2+5, x<=0
g(x)=-sqrt(x-5), x>=5

for f(g(x)) I get x
but for g(f(x)) I get -x

I guess this makes sense looking at the domain and range of f(x), but it bothers me that I'm getting two different values(x and -x).

zarkon · Answer

$$g(f(x))=-\sqrt{f(x)-5}=-\sqrt{x^2+5-5}=-\sqrt{x^2}=-|x|$$

since $x\le 0$ we know that x is negative (or zero)  so |x|=-x

hence $-|x|=-(-x)=x$