Example  SEQ.T.2.  Suppose $10\tan(4x)=5$. Find $x$.
Solution: Since $ A \ne 0$ and $B=4, C=0 $, the equation can be solved with $x =\frac 1B \arctan \frac D A$  where $A=10,B=4,$ and $D=5$. So the key solution is $ x =\frac 14\arctan( \frac 5{10}) =\frac 14 \arctan (\frac 12)$

Comment: We can consider the expression on the left hand side of the equation as a function of $x$ giving $  f(x) = 10\sin(4x)$ . Now the problem can be restated: to find a $x$ where $f(x) = 5$. This problem and its solution can be visualized both on the graph and the mapping diagram for the function $f$. 

For the graph of $f$: Find $y=5$ on the Y axis , then find the point(s) on the graph of $f$ with second coordinate $5$, determine it's first coordinate, and that is the key desired value for $x$.	For the mapping diagram of $f$: Find $y=5$ on the target axis , then find $x$ on the source axis with the function arrow pointing to $5$.
To do this, look for the point where the graph of $f$ intersects the line $y=5$.	To do this,look for the arrow that joins the point on the source axis to the point $y=5$ on the target axis.

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Comments: You can move the (red) point labelled x on the left axis of the mapping diagram to a position where the arrow head points to $f(x) = 5$ and the corresponding point on the graph of $f$  will move to the position where the graph of $f$ crosses the line $ y = 5$.
Check the box and the diagram will show the solution on both the mapping diagram and the graph.
 
You can use the sliders to investigate other examples by $A$ and $B,$ as well as the value of  $D$.

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