When I explain functions to students, among all the information I give them about domain and ranges, I tell them: A function is simply a rule. You put a number in, the function performs some actions on that number according to the rule, and spits out a new number.

So if the function is f(x) = 3x + 2, the rule is "whatever number you put into the function, multiply it by three and add two."

f(1) means "the result when you put 1 into the function." Since the rule is "multiply by 3 and add two," the result can be shown like this:

f(x) = 3x + 2
f(1) = 3(1) + 2 = 3 + 2 = 5

I encourage my students to always write the rule again before performing operations on it. It helps them to visualize that whatever is in the parentheses replaces x in the rule.

We also look at things like f(k + 1). How do you evaluate that?

f(x) = 3x + 2
f(k + 1) = 3(k + 1) + 2 = 3k + 3 + 2 = 3k + 5

Doing several of these types of problems prepares students for composing functions:

If f(x) = 3x + 2 and g(x) = x², what is f(g(x))?

Again, I encourage the students to write the rule out, and then write it with the replacement:

f(x) = 3x + 2
f(g(x)) = 3g(x) + 2
f(g(x)) = 3(x²) + 2 = 3x² + 2

Then they do the reverse: g(f(x))

g(x) = x²
g(f(x)) = f(x)² = (3x + 2)² = 9x² + 12x + 4

The following is a worksheet that gives each of the types of problems shown above.