Example LF.COMP.1

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 $x$.