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Could someone clarify whether this is correct?

If $$D(x) = \frac{5x + 8}{x - 15},$$

would $$D(x^2) = \frac{5x^2 + 8}{x^2 - 15},$$

and $$D(x)^2 = \frac{5x + 8}{x - 15}\times\frac{5x + 8}{x - 15}?$$

Clarification appreciated!

Calvin Khor
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    Yes, that‘s correct. – Qi Zhu Aug 24 '20 at 15:11
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    Assuming that $D(x)$ is notation for a function from $x$ to $D(x)$, then yes. – mathreadler Aug 24 '20 at 15:12
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    You may also see it written $D(x)^2=D^2(x)=(D(x))^2$, e.g. for the trig functions you often see things like $\sin^2(x)$ to mean $(\sin x)^2$, for example. But beware of "inverse functions" as these are sometimes written $f^{-1}(x)$ and, unless otherwise stated, this is not intended to mean $1/f(x)$, e.g. $\sin^{-1}(x)$ is the inverse sine function, not $1/\sin(x)$. – pshmath0 Aug 24 '20 at 15:16

3 Answers3

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The parenthesis behind a function identifier isolate its argument. So

$$D(x^2)$$ perforce denotes the function $D$ evaluated at $x^2$.

On the other hand,

$$D(x)^2$$ cannot be interpreted as $$D((x)^2)$$ because the inner parenthesis would be superfluous and the outer ones missing. Hence you have the choice between

  • meaningless, or

  • $(D(x))^2$.

An alternative notation is $D^2(x)$, though in some rare cases (which you will be aware of by context), this designates $D(D(x))$.

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Yes, you are completely right. However, it might be worth noting a few other things:

• Trigonometric functions do not follow this pattern. $\sin^2(x)$ means $(\sin x)^2$, not $\sin(\sin(x))$.

• Some notations are more readable than others. In my opinion, $(\log x)^2$ is preferable to $\log(x)^2$, though they both unambiguously mean the same thing. (Incidentally, some authors adopt a trigonometric-type convention for logarithms, which is awful!)

Joe
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  • What’s so wrong with $\log^2x$? – gen-ℤ ready to perish Aug 24 '20 at 16:35
  • @gen-zreadytoperish Ultimately, this is just my personal preference, but I think $\log ^2 x$ goes against the 'natural' interpretation of the exponent. If $x^2$ means $xx$, then it would seem logical that $\log^2 x$ would mean $\log \log(x)$—that is, $\log(\log(x))$. Furthermore, it seems unnecessary to contrive a new notation when $(\log x)^2$ already has a clear meaning. – Joe Aug 28 '20 at 20:59
  • I see what you’re saying. I got used to the squared meaning after years dealing with $\sin^2$. As to the other point, the benefit is omitting the parentheses. – gen-ℤ ready to perish Aug 28 '20 at 21:00
  • @gen-zreadytoperish Very true. At times, I do see the benefit in getting rid of the clutter of the parentheses. It's fascinating how the notation you use shapes how you think about mathematics. I often have a different intuitive conception of the derivative if it is written as $f'(x)$ rather than $dy/dx$. – Joe Aug 28 '20 at 21:04
  • @gen-ℤreadytoperish On an entirely unrelated note, I love how you have changed your username. I suppose we can find comfort in the fact that if generation Z does perish, $\mathbb{Z}$ will be unaffected! – Joe Sep 04 '20 at 21:03
  • I feel so special that someone noticed the change! lmao. I made it two nights ago. It surprises me that none of the moderators have pressed me about it to be honest, even after my account was suspended for a week because I had profanity in my bio. – gen-ℤ ready to perish Sep 04 '20 at 22:55
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The first two are correct because this is a function D with an independent variable $x$ ( $D$ of $x$). However, the last one is not proper notation. If you're gonna put a square outside of $x$ then it means you're squaring the whole left side $(D(x))^2$. For trig functions it's different.

user577215664
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Cornel Stoica
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