A one-to-one function
A function that is not one-to-one
Constant function
Identity function
Not a function
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Overthinking my teaching 
AWESOME. ^_^
What he said. I’m not sure how evocative it is as a standalone image, but it’s quite clever with just a bit of thought.
Have you ever used this with your students? If so, how did they react?
The vending machine metaphor actually resonates with my College Algebra students pretty well every semester. Several students have come back to me and mentioned using vending machines to think about functions in later courses.
Of course, we deal with the formal definition of function, and with more abstract imagery as well. But I really do think we ignore this sort of informal imagery at our (and our students’) peril. They’ll form images whether we support that process or not.
And I’m agreed that these don’t work as standalone images. You do need to think them through, and discuss them.
Plus they’re fun.
Oh, by the way. I don’t test my students on their ability to use the vending machine metaphor. The standard is about functions. Vending machines are a route to the standard, but are not the standard.
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Christopher, how do you expand on this with the students?
What’s the domain and range of your “identity” function?
Values. Domain is {1,5}. Range is {1,5}. Function is {(1,1), (5,5)}.
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I love this. I normally usee the example of a toaster. The white bread is the input; the toaster is the FUNction rule; the toast the output. NOW, what type of FUNction is the toast? If I put in rye bread, I still get toast out, just like if I toasted wheat.
Thanks for the ideas and for keeping me thinking how I approach the topic of FUNctions w/ my students.
I love the different vending machines idea.
Oh, one thing I try to do is ask my students what do they see…in hopes of getting to ‘sameness’. Maybe there’s something they see I don’t see.
This is great. Any idea on how we use this metaphor with an onto function?
Love thus applicatin of types of functions.
The discussion broadens when you describe the input (x-value) as the amount of money inserted.