I know, loosely speaking, if we can define a function $f$ in term of

begin{align}

&f(0,vec{x})=g(vec{x})\

&f(n+1,vec{x})=h(f(n),n,vec{x})

end{align}

where functions $g,h$ are primitive recursive. Then $f$ is primitive recursive.

However, what it means to show a function is primitive recursive by induction?

I had read above explaination on page 93 on book $textit{Computability}$ by Epstein and Carnielli, but still I’m not sure if I got the idea. Could someone provide some examples about how a inductive definition shows a function is primitive recursive?