How would you explain function notation to a friend?

Many students seem puzzled by function notation well after it has been introduced, and ask “why can’t we just write

y = 3x instead of

f(x) = 3x ?”. To motivate the use of function notation and improve understanding, I advocate using multi-variable functions instead of single-variable functions in introducing this notation. My introduction usually proceeds something like this:

Step-by-step explanation:

option 1

data plot with   3 dots over 33, 1 dot over 34, 2 dots over 35, no dot over 36, 1 dot over 37, 2 dots over 38, 4 dots over 39, and 2 dot over 40.

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step-by-step explanation: