Functions without inverses

Here are some problems that could get at the idea that not every function has an inverse function:

Number Tricks: “Pick a number between 1 and 20. Subtract 10. Square the result, Add 7.” Then, list everyone’s starting and ending numbers. Anyone get the same ending number from different starting numbers? Why? Then, swap squaring with cubing. Try it again. What happens? Number tricks are also helpful for constructing inverses.

Categorizing Graphs: “Here are 5 different graphs. Put them into two categories.” They’ll do what they do, but we’ll focus in on one particular way of categorizing the graphs — those that loop back up/down, like y = x^{2} and those that don’t. What’s the difference between the tables of those graphs?

Piece-wise functions: f(x) = { x+1   if x>2, and x-1  if x<3 . Then ask them to make a table for this function. What do you come up with?

What’s the input? Maybe just introduce some notation like input_{f}(x) and then ask something like, “Given that f(x) = x^{2}, what is output_{f}(5)? What is input_{f}(9)?”

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