Here is the question I need to do:
Consider the following functions: $$\begin{align}f:\Bbb N\to\Bbb B&\text{ defined by }f(x)=x>8;\\g:\Bbb N\to\Bbb N&\text{ defined by }g(x)=(x\cdot 3) - 7;\\h:\Bbb N\to\Bbb N&\text{ defined by } h(x)=(x-2) + (x+5).\end{align}$$For each function composition below, say whether the composition is valid or not, and for each valid function composition define what the resulting function does. $$\begin{align}\textbf{i)}\,\,f\circ g\qquad&\textbf{ii)}\,\,f\circ f & \textbf{iii)}\,\,g\circ h\qquad\\ \textbf{iv)}\,\,h\circ h\qquad&\textbf{v)}\,\, h\circ f\circ g &\textbf{vi)}\,\,f\circ h\circ g\\ \end{align}$$
Before I do the rest could you check if I'm doing i) correct? Thanks!
It is valid since $\mathop{\textrm{cod}} g=\mathop{\textrm{dom}} f, \mathbf N=\mathbf N$
$f\circ g(x)=f(g(x))=f((x\cdot 3) -7))=(x\cdot 3) -7 > 8 = 3x -7>8=3x>15=x>5$