Verify associativity for the following three mappings

Question:

Verify associativity for the following three mappings: $f: N \rightarrow Z_{0}$ (the set of non-zero integers), $g: Z_{0} \rightarrow Q$ and $h: Q \rightarrow R$ given by $f(x)=2 x, g(x)=1 / x$ and $h(x)=e^{x}$.

Solution:

Given that $f: N \rightarrow Z_{0}, g: Z_{0} \rightarrow Q$ and $h: Q \rightarrow R$.

$g \circ f: N \rightarrow Q$ and $h o g: Z_{0} \rightarrow R$

$\Rightarrow h \circ(g o f): N \rightarrow R$ and (hog) of $N \rightarrow R$

So, both have the same domains.

$(g o f)(x)=g(f(x))=g(2 x)=\frac{1}{2 x} \quad \ldots(1)$

$($ hog $)(x)=h(g(x))=h\left(\frac{1}{x}\right)=e^{\frac{1}{x}} \quad \ldots(2)$

Now,

$(h o(g o f))(x)=h((g o f)(x))=h\left(\frac{1}{2 x}\right)=e^{\frac{1}{2 x}}$                    $[$ from (1) $]$

$((\operatorname{hog}) \circ f)(x)=(\operatorname{hog})(f(x))=(\operatorname{hog})(2 x)=e^{\frac{1}{2 x}}$               $[$ from (2) $]$

$\Rightarrow(h o(g o f))(x)=((h o g) o f)(x), \forall x \in N$

So, $h o(g o f)=(h o g) o f$

Hence, the associative property has been verified.