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Can I call functions like $\sin x , \tan^{-1} x , e^ x $ etc composite functions too?

If not, then what do I call them? I mean taking individual names like exponential function, sine function etc doesn't seem convenient. What is correct term for this? How do I refer to these functions?

To be more specific. Eg : the natural numbers greater than 1 that are not prime are called composite numbers. And those that are not composite are prime numbers. It either prime or composite.

So how do I refer to functions that are not composite functions? Do we have a word for them? Like Prime function or something?

William
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    Every function can be written as a composite function. $f(x)=x$ can be written as $g\circ h (x)$ where $h(x)=x+1$ and $g(x)=x-1$ for example. – lulu Jul 14 '18 at 10:03
  • @lulu so you mean EVERY function is a composite function? – William Jul 14 '18 at 10:04
  • Exactly. If $F(x)$ is any function, let $H(x)=F(x)+1$ and $G(x)=x-1$. Then $F=G\circ H$. – lulu Jul 14 '18 at 10:04
  • On set $X$ is the identity function $\mathsf{id}_X$ prescribed by $x\mapsto x$. For every function $f:X\to Y$ we have $f=f\circ\mathsf{id}_X=\mathsf{id}_Y\circ f$. – drhab Jul 14 '18 at 10:09
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    Nevertheless I have understanding for your motivation to ask this. See this question of myself in another context. I would rather speak of irreducible functions. – drhab Jul 14 '18 at 10:16
  • I would say that the phrase composite function is misleading because "is composite" is not an interesting property: Every function is the composition of two functions, as others have shown. In fact I've never seen this terminology used outside of a precalculus course. So I suggest you either avoid using this term, or think of it as a nontechnical term without a precise meaning. – Yakov Shklarov Jul 14 '18 at 18:47

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Some functions are obviously composite, such as $$f(x)=e^{\sin x}$$ or $$f(x)= \tan^ {-1} (x^2+1)$$

Other functions can be considered as composite functions.

For example the identity function $id(x)=x$ could be considered as$$ id(x)=fof^{-1}(x)$$ for any bijective function $f(x).$

Mohammad Riazi-Kermani
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A function $f:\>A\to B$ is a function, namely a subset $f\subset A\times B$ satisfying certain conditions.

If a function $f:\>A\to B$ is given, and you can find a set $C$, together with functions $g:\>A\to C$ and $h:\>C\to B$ such that $f=h\circ g$ (where $\circ$ is properly defined) then $f$ is called the composition of $g$ with $h$.

Of course you can collect a list of functions occurring in calculus 101, like $$x\mapsto x,\quad x\mapsto x^2,\quad \exp,\quad\log,\quad\sin,\quad x\mapsto\sqrt{x},\quad{\rm etc.}\ ,$$ and call them basic functions of calculus, but the members of this list are up to personal taste. In any case, there is no such thing as a prime function with respect to composition.

Christian Blatter
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