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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: Stroodle on June 18, 2009, 01:06:35 am

Title: Range of composite functions question
Post by: Stroodle on June 18, 2009, 01:06:35 am: Hi.

If:
$f:R\rightarrow R,\;f(x)=x^2-4$
and
$g:R^+\cup\left\{0\right\}\rightarrow R,\;g(x)=\sqrt x$

Then what is the range of $f\circ g$ and why?

Thanks for your help.
Title: Re: Range of composite functions question
Post by: ilovevce on June 18, 2009, 03:19:50 am: To find the range $f\circ g$ we must first find the domain of this function.

The domain of $f\circ g$ is equal to the domain of $g$ , as long as the range of $g$ is a subset of the domain of $f$ . If not, the range of $g$ must be restricted.

For this question, the domain of $g$ is $[0,\infty)$ and the domain of $f$ is $R$

The range of $g$ is $[0,\infty)$ . These values of the range of $g$ are the values that will be plugged in to $f$ .

We can check and see that the range of $g$ fits in the domain of $f$ . It does. So we don't have to further restrict the domain of $f\circ g$ .

$\therefore dom f\circ g = dom g = [0,\infty)$

The range that corresponds to this domain is $ran f\circ g = [-4,\infty)$
Title: Re: Range of composite functions question
Post by: Stroodle on June 18, 2009, 10:01:07 am: Awesome. Thanks for that
Title: Re: Range of composite functions question
Post by: Toothpaste on June 18, 2009, 12:41:38 pm: Would this be of help too?
http://vcenotes.com/forum/index.php/topic,1283.msg18391.html#msg18391