Misconceptions and Difficulties in the Understanding
Of the Function Concept

The student whose concept of a function is incorrect or imprecisely mastered often:

1.   Believes that all functions are linear.
 
2.   Believes that all functions are one-to-one.
 
3.   Believes that all functions in real life are one-to-one.
 
4.   Has difficulty making examples of functions in real life that are non-linear, not one-to-one, constant, and/or whose graphs are disconnected.

5.  Has an overly restricted view of the forms that graphs of functions can take.
 
6.  Believes the graph of a function should be "reasonable," that is, it should be "regular"--not have any angles.
 
7.  Believes that relations represented by graphs that are disconnected are not functions.  (This difficulty/belief may not appear when the algebraic representation is given.)
 
8.   Believes that changing the dummy variable in the algebraic representation changes the function.  That is, believes {(x,f(x)) | f(x)=x2+3} and {(t,f(t)) | f(t)=t2+3} are different functions.
 
9.   Fails to realize that all ordered pairs on the graph satisfy the algebraic equation defining the function.
 
10.   Believes constant functions are not possible.  Equivalently, the student believes if the image set is a single value, then it isn't a function, or if all the second coordinates are the same, then it isn't a function.

11.  Has the expectation regarding a function that is embodies causality.
 
12.   Has difficulty with the terms preimage, image, (preimage, image) pair, domain, range, image set.

13.  Has difficulty moving from one representation of a function to another.

14.  May have a variety of unrelated methods for determining if  something is a function,  depending on the representation.  There may be no connections between methods  and no connections to the ("formal") definition of a function. 
   
15.   Has difficulty determining the preimage and image for points that are graphed.
 
16.  Has difficulty determining the domain and image set for a function (The function may be given in algebraic or graphical form.).
 
17.  Has difficulty with points (ordered pairs) that are on the axes.
 
18.  Often ignores the domain and range of the function.
 
19.  Has difficulty distinguishing between the image set and the range of a function.
 
20.  Fails to appreciate functions defined piecewise.
 
21.  Has difficulty with functions defined piecewise.
 
22.  Has difficulty determining the values of the domain for which the function values are increasing (decreasing).  In fact, the student may have difficulty understanding what the directive means!
 
23.  Has difficulty determining the values of the domain for which the function values are a maximum (minimum).  Again, the student may have difficulty understanding what the directive means.
 
24.  Has a tendency to "see" only the marked points on a graph where a few points are plotted, then a curve or line is drawn through them.
  
25.  Restricts his/her focus to an individual point or group of points as opposed to the more global features of the graph.

26.  Has difficulty reporting a set of numbers (interval such as x>4, or -2#x#2).

27.  Interprets a graph of a situation as a picture of the actual situation!  

28.  Confounds slope and height, when interpreting and constructing graphs.

29. Confounds height and difference in height, when interpreting and constructing graphs.  

30. Confounds slope and curvature, when interpreting and constructing graphs.