Help with composite functions? Suppose that $u$ and $w$ are defined as follows: 
$u(x) = x^2 + 9$
$w(x) = \sqrt{x + 8}$
What is: 
$(u \circ w)(8) = $
$(w \circ u)(8) = $
I missed this in math class. Any help?
 A: Look at the expression for $u(x)$. When we write $u(w(8))$, it means to use $w(8)$ everywhere that had "$x$" in the expression for $u(x)$. However, $w(8)$ is just a number, you can calculate separately using the expression for $w(x)$. And finally, $u(w(8))$ and $w(u(8))$ need not be equal. 
A: $$w\circ u(8)=\sqrt{\left(8^2+9\right)+8}=9$$
$$u\circ w(8)=\left(\sqrt{8+8} \right)^2+9=25$$
Note that as Ivo says, they aren't equal.
A: When the function isn’t too complicated, it may help to express it in words. So, your $u$ is “square the input, and then add $9$, to get your final output”. And your function $w$ is “add $8$ to your input, and then take the square root to get your final output”. And I’m sure you know that $u\circ w$ means to perform the $w$ process first, and then, using your output from $w$ as input to $u$, perform the $u$ process. That is exactly what @olive euler has done.
A: $(u\circ w)$ is the composition of function $u$ over $w$.  
It means: apply function $u$ to the result applying function $w$.  That is all.
$$(u\circ w)(x) = u(w(x))$$
And conversely: $(w\circ u)(x) = w(u(x))$
