Solution of composition of function

Question

In a book I saw a question along with solution

The question is

Let f,g,h be function from R to R , then show that (f+g)oh = (foh).(goh)

But when I saw the solution i got confused , they have cancelled the term x from both sides how ?

Answer 1

Remember that two functions are identical if they have the same domain (which, in your case, they do) and the same values.

So, all you have to prove is that for every $x$ in $\mathbb R$, the value $((f+h)\circ h)(x)$ is the same as the value $(f\circ g + g\circ h)(x)$.

There is no "canceling of $x$" anywhere. The proof is a typical proof of a statmenet of the form "for all $x$, $P(x)$ is true":

1. Let $x$ be an arbitrary value of $x\in\mathbb R$.
2. Then, $((f+g)\circ h)(x) = (f+g)(h(x))$ (by definition of compositums)
3. $=f(h(x)) + g(h(x))$ (by the definition of what a sum of functions is)
4. $=(f\circ h)(x) + (g\circ h)(x)$ (again, definition of compositums, used in the other direction)
5. $=((f\circ h) + (g\circ h))(x)$ (by the definition of the sum of functions, used in the other direction)
6. Therefore, we have proven for this value of $x$ that $((f+g)\circ h)(x)=((f\circ h) + (g\circ h))(x)$.
7. Because $x$ was arbitrary, we have proven that the above holds for all values of $x$.
8. Therefore, $(f+g)\circ h=(f\circ h) + (g\circ h)$

Answer 2

I hope this will clear up some misunderstanding about what you call "cancelling $x$".

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If you prescribe functions $u:\mathbb R\to\mathbb R$ by $x\mapsto\sin x$ and $v:\mathbb R\to\mathbb R$ by $x\mapsto\cos(x-\frac12\pi)$ then: $$u=\{\langle x,\sin x\rangle\mid x\in\mathbb R\}=\{\langle x,\cos(x-\frac12\pi)\rangle\mid x\in\mathbb R\}=v$$

Sometimes this is expressed as $u(x)=v(x)$ where $x$ is looked at as a variable ranging over domain $\mathbb R$. This expression is tricky in the sense that $f(x)=g(x)$ can also be used to express that eventually distinct functions $f$ and $g$ take the same value at $x$. In that interpretation $x$ must not be looked at as a variable, but as fixed element of the domain.