Ashli

@mythagon.bsky.social

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When it comes to inverse functions, the expectations are modest, requiring only that students solve equations of the form f (x)=c. The point is to provide an informal sense of determining the input when the output is known. Much of this work can be done with specific values of c. Eventually, some generality is warranted. For example, if f(x)=2*x^3, then solving f(x)= c leads to x = (c/2)^(1/3), which is the general formula for finding an input from a specific output, c, for this function, f. At this point, students need neither the notation nor the formal language of inverse functions, but only the idea of “going backwards” from output to input. This can be interpreted for a table and graph of the function under examination. Correspondences between equations giving specific values of the functions, table entries, and points on the graph can be noted (MP.1). And although not required in the standard, it is reasonable to include, for comparison, a few examples where the input cannot be uniquely determined from the output. For example, if g(x) = x^2, then g(x)= 5 has two solutions, x = +/- sqrt{5}.
Students also formalize the notion of a transformation as a function from the plane to itself. When the transformation is a rigid motion (a translation, rotation, or reflection) it is useful to represent it using transparencies because two copies of the plane are represented, one by the piece of paper and one by the transparency. These correspond to the domain and range of the transformation, and emphasize that the transformation acts on the entire plane, taking each point to another point. The fact that rigid motions preserve distance and angle is clearly represented because the transparency is not torn or distorted.
Initially, students view rigid motions as operations on figures (“transformations in the plane”). Later, students come to understand that it is not the figure that is translated, rotated, or reflected, it is the plane that is moved, carrying the figure along with it. Students start thinking, not of moving one figure onto another, but of moving the plane so that the first figure lands on the second (“transformations of the plane”) without moving the coordinate grid. This change in perspective is makes it possible to describe the effect of a given rigid motion on any point in the plane.