What's the difference between $f(x)=\sqrt{x^2+9}$ and $k(x^2+9)=\sqrt{x^2+9}$? Let's say we 've got a function $f(x)=\sqrt{x^2+9}$, which is a composite function. $f(x)=\sqrt{g(x)}$ and $g(x)=x^2+9$.
When we have a function like $h(x)=x$, we are allowed to set $x$ to $x+9$ and have $h(x+9)=x+9$.
So why do we need $g(x)$ and can't just set $x=x^2+9$, which with a function like $k(x)=\sqrt{x}$ leads to $k(x^2+9)=\sqrt{x^2+9}$ (same as f(x) above)? 
Where's the difference between these two ($f(x),k(x)$)? Is $x^2+9$ even a valid argument for $k(x)$?

$f(y)$ is a function which maps $y$ to $\sqrt{y^2+9}$, and you've fed $y=x$ into it to get your first function.
$k(y)$ is a function which maps $y$ to $\sqrt y$, and you've fed $y=x^2+9$ into it to get your second function.
