My brother has not taken a math class in $10-15$ years. He is studying for the GRE so I have been teaching him a chapter or two from my precalculus book. So far, he has learned (and excelled at) basic algebra skills such as factoring quadratics, solving equations and inequalities in one variable, graphing lines, etc. But now the section on functions is coming up.
I am having a hard time finding a good balance between between a rigorous and an informal definition of a function. If I make it too rigorous (a function is a subset of $A \times B$ with blah blah blah propreties), the topic will seem useless and very separated from the mathematics he has done so far. If I make it too informal (a function is just a rule that assigns members from $A$ to members of $B$, with certain restrictions), there will be a lot of confusion ("So is the function the actual arrows from $A$ to $B$?" "Is the equation $f(x) = 2x$ the function?" Or is $f$ the function?" "Is $f(x)$ the function?" "Is $2x$ a function?")
My current plan is to get the best of both worlds, but I would like to know if there is a better way to teach it, and I would also like to know what I should teach after going over the definition.
A function from $A$ to $B$ is a triplet $A, B, f$ where $A, B$ are sets and $f$ is some sort of rule that assigns a unique value $b \in B$ to each $a \in A$. The rule coult be a bunch of arrows, an equation, among other things.
Using this method I think I will still get some of the confused questions mentioned above, but I think using this definition I can defend the position that the function is the triplet $A, B, f$ and the book is being lazy if they say something like "Find the zeroes of the function $f(x) = 2x$". I would still like some advide about answering the "confused questions" though if you have it.
I still think that my brother will find questions such as "Find the zeroes of $f(x) = 2x$" as useless and fluffed up ways of saying "Solve $2x = 0$. How can I alleviate his concerns?