# Homework Help: Find the compositions of these functions?

1. Mar 29, 2010

### 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 ?

2. Mar 29, 2010

### justsof

Re: function

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

3. Mar 29, 2010

### Fightfish

Re: function

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

4. Mar 30, 2010

### thereddevils

Re: function

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

5. Mar 30, 2010

### Martin Rattigan

Re: function

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.

6. Mar 30, 2010

### thereddevils

Re: function

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.

7. Mar 30, 2010

### Martin Rattigan

Re: function

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

8. Mar 30, 2010

### thereddevils

Re: function

thanks again Martin !