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Composite functions:

Let $g(x)=3x+1$. Express $g^2(x)$ in terms of $x$.

I have tried squaring $3x+1$, but I think I'm going about it the wrong way... Help is much appreciated. Thanks.

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The expression $g^2(x)$ means $g(g(x))$. To write this out with your specific $g$, we get $$g(g(x)) = 3g(x) + 1$$Now we substitute again. $$g(g(x)) = 3g(x) + 1 = 3(3x+1) + 1$$Then simplify that.

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    I suppose you’re right, but the notation $g^2$ is ambiguous without some prior explanation. It often means $g\circ g$, but it can also mean $g\cdot g$. Like $\sin^2(x)$. – 2017-02-10
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    @Lubi You're right that it is somewhat ambiguous, because of exactly the expression $\sin^2(x)$. However, using $g^2$ for $g \circ g$ is much more common, and I would even argue that using $\sin^2(x)$ to mean $(\sin x)^2$ is bad notation, simply because it causes this confusion. – 2017-02-10
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    I’m with you on the badness of “$\sin^2$”. I still don’t know what it means in analytic number theory to write $\log^2$ — usually, they write $\log\log$, so I guess the other really is $(\log x)^2$. – 2017-02-10