Function is defined in many textbooks as a correspondence relationship from set X to Y such that for every x (element of X), there is one and only one y value in Y. Definitions are important to know but in the case of function, the only time students will ever use the definition of function as correspondence is when the question is “Which of the following represents a function?”. I think it would be more useful for students to understand function as a dependence/covariational relationship first than for them to understand function as a correspondence relationship. The latter can come much later. In dependence/co-variational relationship “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (Sfard, 1991, p. 15)
The concept of change and describing change is a fundamental idea students should learn about functions. Change, properties, and representations. These are the big ‘ideas’ or components we should emphasize when we teach functions of any kind – polynomial, exponential, logarithmic, etc. Answer the following questions to get a sense of what I mean.
1. Which equation shows the fastest change in y when x takes values from 1 to 5?
A. y = 4x2 B. y = -2x2 C. y = x2 + 10 D. y = 6x2 – 5
2. Point P moves along the graph of y = 5x2, at which point will it cross the line y = 5?
A. (5, 0) and (0,5) B. (-5, 0) and (0,-5) C. (1, 5) and (-1, 5) D. (5, 1) and (5,-1)
A. h(x) = x3 + 1 B. h(x) = x3 – 1
C. h(x) = 2x3 + 1 D. h(x) = 2x3 + 4
4. The zeros of the cubic function P are 0, 1, 2. Which of the following may be the equation of the function P(x)?
A. P(x) = x(x+1)(x+2) B. P(x) = x(x-1)(x-2) C. P(x) = x3 – x2 D. P(x) = 2x3 – x2 – 1
5. Cubes are made from unit cubes. The outer faces of the bigger cube are then painted. The cube grows to up to side 10 units.
The length of the side of the cube vs the number of unit cubes painted on one face only can be described by which polynomial function?
A. Constant B. Linear C. Quadratic D. Cubic function
Item #1 requires understanding of change and item #5 requires understanding of the varying quantities and of course the family of polynomial functions.
Of course we cannot learn a math idea unless we can represent them. Functions can be represented by a graph, an equation, a table of values or ordered pair, mapping diagram, etc. An understanding of function requires an understanding of this concept in these different representations and how a change in one representation is reflected in other representations. Items #2 and #3 are examples of questions assessing understanding of the link between graphs and equations.
Another fundamental idea about function or any mathematical concept for that matter are the properties of the concept. In teaching the zeroes of a function for example, students are taught to find the zeroes given the equation or graph. One way to assess that they really understand it is to do it the other way around. Given the zeroes, find the equation. An example of an assessment item is item #4.
You may also want to read How to assess understanding of function in equation form and Teaching the concept of function.