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\$
• Click on the arrow button in the figure to start the animation.
• The button will change to a "pause" button, which can be used to halt the animation.
Notice how the point on the graph is paired with the points and arrow on the mapping diagram.
• Check the box to see the mapping diagram and graph of \$f\$ .
• Check the box to see the mapping diagram and graph of \$f \circ g\$.
• Select the point \$x\$ in the mapping diagram to move the point by mouse or use the scroll up or down key  to move by increments of 1.

INsert GeoGebra Here;
 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\$.