# computability – How to show a function is primitive recursive by induction?

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?