Find the compositions of these functions?

[thereddevils]

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 ?

 

[justsof]

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

 

[Fightfish]

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

 

[thereddevils]

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

 

[Martin Rattigan]

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.

 

[thereddevils]

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.

 

[Martin Rattigan]

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

 

[thereddevils]

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

thanks again Martin !