# What are algebra functions and how do they work?

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Trying to explain to my high schooler in simple terms so he (and I) can get it. He sees f(x) and just shuts down.

Thanks for all the answers. I will try these. He panics when he has math because he was called “stupid” and put in a slow learners class in elementary school.

In: Mathematics It’s a machine or tool with an input and an output. The math just determines what happens in the middle 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. Algebra is simply exchanging numbers for letters or symbols, so that rather than being a fixed number they become variable. A complex algebra equation will explain the relationship between many different algebraic numbers tied together by an equation. 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. 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. 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