I have a problem with the following exercise:
As we saw in this section, the iteration process can be carried out either algebraically or graphically. Composition of functions can also be represented graphically. The accompanying figure shows the graphs of two functions $f$ and $g$, the line $y=x$, and an input $a$. Use the figure and the ideas of this section to discover and then explain how to find the point $(a, g[f(a)])$ on the graph of the composite function $g\circ f$
I don't know how! I need some help. I tried for more than an hour without success.
First, draw a vertical line from $a$ (at the $x$-axis) to $f(x)$, marking the point $(a,f(a))$. Now draw a horizontal line from $f(a)$ to the line $y=x$, marking the point $(f(a),f(a))$. From here draw another vertical line to $g(x)$, marking the final point $(f(a),g(f(a))$. Finally, find the intersection of the lines $x=a$ and $y=g(f(a))$ so that you end up with the desired point $(a,g(f(a)))$.