\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2015e.00724}
\itemau{Esty, Warren}
\itemti{Teaching about inverse functions.}
\itemso{AMATYC Rev. 26, No. 2, 4-10 (2005).}
\itemab
Summary: In their sections on inverses most precalculus texts emphasize an algorithm for finding $f^{-1}$ given $f$. However, inspection of precalculus and calculus texts shows that students will never again use the algorithm, which suggests the textbook emphasis may be misplaced. Inverses appear primarily when equations need to be solved, which suggests instruction about inverses should emphasize their use in solving the equation ``$f(x)=c$." Instruction, and the algorithm used, should take advantage of the possibility of perfectly paralleling the process for solving ``$f(x)=y$" for $x$ (not solving ``$f(y)=x$" for $y$). Switching letters after solving, rather than before solving, preserves the parallel. When $f$ is not one-to-one (such as $f(x)=x^2$ or $f(x)=\sin x$), students frequently fail to find the second solution. By discussing inverses in terms of solutions to ``$f(x)=c$," this difficulty is naturally addressed. Furthermore, the terms one-to-one and range have natural definitions in this context and the Horizontal Line Test is also natural. The algorithm for finding inverses and the associated terminology can best aid in proper conceptual development if they focus on the primary context -- solving ``$f(x)=c$" for $x$.
\itemrv{~}
\itemcc{I20}
\itemut{inverse of a function; approach; algorithms; solving equations; calculus; elementary algebra; concept formation; non-bijective functions; range}
\itemli{}
\end