Example  1.   $ f(x) = 2x + 1; g(x) = 3x + 2$.

Dynamic Views for Composition of Linear Functions $f \circ g$:Tables and Mapping Diagrams
$f(x)= m_fx + b_f$ and $g(x) = m_gx+b$

$(f \circ g)(x) = f(g(x)) =f(3x+2)= 2*(3x+2) + 1 = (6x +4) +1= 6x + 5$
For the graph of $f$: Find $y=20$ on the Y axis , then find the point on the graph of $f$ with second coordinate $20$, determine it's first coordinate, $6$, and that is the desired value for $x$. For the mapping diagram of $f$: Find $y=20$ on the target axis , then find $x$ on the source axis with the function arrow pointing to $20$.

To do this, look for the point where the line $y=20$ intersects the line that is the graph of $f$ To do this,  find the focus point of $f$, $F= [5, -10]$ on the mapping diagram. [Use Geogebra?] Draw the line through F and the point $y=20$, to find the point of intersection of this line with the X axis, $x=6$, which is the desired value for $x$.