Difference between $D(x^2)$ and $D(x)^2$?

Question

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!

Answer 1

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!)

Answer 2

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 $D(D(x))$.

Answer 3

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.