# Find the compositions of these functions?

Given two functions , f(x)=x^2+3 , where x is real , g(x)=|x|-5 , x is real , find gf(x).

i found gf(x)=x^2-2

is it true that the range of gf(x) is the same as the range of g(x) ? If so,

the range of g(x) is [-5 , infinity) and the range of gf(x) is [-2 , infinity)

why arent they the same ?

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I think everything is correct, why would you think the range of g(f(x)) is the same as the range of g(x)?

I'll pose the reverse question: why should they be the same? g(x) and gf(x) are different functions.

Take a look at the diagram i attached . Isn't that the image of gf(x) the same as the image function g(x) ?

The range of $f$ needn't be the whole of the domain of $g$ so some of the images under $g$ may not occur in $gf$. You only have $range(gf)\subseteq range(g)$. In fact you could say they're not equal because you have yourself provided a counterexample.

The range of $f$ needn't be the whole of the domain of $g$ so some of the images under $g$ may not occur in $gf$. You only have $range(gf)\subseteq range(g)$. In fact you could say they're not equal because you have yourself provided a counterexample.
thanks Martin , how about the domains , is the domain of f(x) the same as domain gf(x) because both of them started from the same set or it needn't also be the same in this case.

Yes, $dom(gf)=dom(f)$, assuming you only define the composition $gf$ when $range(f)\subseteq dom(g)$.

Yes, $dom(gf)=dom(f)$, assuming you only define the composition $gf$ when $range(f)\subseteq dom(g)$.
thanks again Martin !