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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?

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4 Answers 4

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.

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$$w\circ u(8)=\sqrt{\left(8^2+9\right)+8}=9$$ $$u\circ w(8)=\left(\sqrt{8+8} \right)^2+9=25$$ Note that as Ivo says, they aren't equal.

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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.

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$(u\circ w)$ 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: $(w\circ u)(x) = w(u(x))$

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