The term “function” just means “equation based on or dependent on a variable” you can substitute a different variable. For example if “y” is a function of “x” and that function is “cos(x)+isin(x)” then f(x)=cos(x)+isin(x) and y=f(x) so y=cos(x)+isin(x).
The main purpose of learning to think in terms of f(x) =mx+b rather than y=mx+b is because f(x) works easier in calculus.
Your high schooler’s stumbling block just seems to be the notation?
Taking a step back, to motivate the notation, we can start with simple intuitive examples. Suppose I mail people twice as much money as they mail to me. You mail me $2, I mail you back $4. If you mail me $2.50, I mail you back $5.
You can send any amount, so let’s use a placeholder *x* to represent the amount.
My job is to double that amount, which can now be expressed as 2*x*.
That’s my job. That’s my “function”, here. Right? My “function” is to receive *x*, and return 2*x*. That’s one way to help remember the terminology.
“My function expects me to receive *x* from you” can be written in math notation as *f*(*x*) . It’s like a drawing of me holding *x* in my arms.
And what do I return to you? In my example, we said it’s 2*x*.
So, in full math notation, we can write this example as *f*(*x*) = 2*x*
That’s one relation. All kinds of relations can be expressed through functions, including trigonometric relations and more.
It’s like a magic box that does a specific thing, and let’s you put anything in, and outputs another thing that has that initial thing applied to it.
Let’s say f(x) = toaster
If I do f(bread), I’m inserting bread into the toaster, and I’m going to get toast
If I do f(bagel), I’m inserting a bagel into the tater, and I’ll get a toasted bagel
You could think about the “cloud to butt” plugin on a web browser that replaces every instance of the word “cloud” with “butt”
f(cloud) = this is a sentence with the word cloud in it
f(butt) = this is a sentence with the word butt in it
I hope that the non-math examples help.
You know how variables represent unknown numbers? Functions represent unknown operations (addition, subtraction, multiplication etc.)
Taking f(x) is taking x and applying a “mystery operation,” such as f(x)=3x. In this case, the mystery operation is multiplying by three. As is usual, I blame mathematicians for coming up with confusing notation that doesn’t make a lot of sense.
I would explain this using a diagram. It’s like an input, a “box”, and an output. To be explicit:
For f(x) = x + 2
Input | Function box | Output
x = 1 | 1 + 2 | 3
x = 12 | 12 + 2 | 14
So when you see f(x), you should be thinking about it in terms of of the above.
I would also drive the point home that if f(x) = x + 2 … then f(y) = y +2 … it’s just a placeholder for a value to be put in and taken out.
You can extend it as well because “f” is just a signifier that it’s a function. You could equally say g(y) = y + 2, where g is the function instead.
Finally, you can combine them together like f(x) = x^2 and g(y) = y + 2. Then g( f(x) ) does the same thing. The input uses the function to generate an output so:
– g( f(x) ) = f(x) + 2 = x^2 + 2
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