What is a Function?

A function is a set of instructions that you perform on an input or on multiple inputs in order to produce an output. Whenever you are given a function, you should identify what the input is (inputs are) and figure out what instructions you are being asked to perform on that input (those inputs). It is important that you not get hung up on what specific variable is being used (or variables are being used); the more important thing to do is to understand what instructions/operations are being asked to be performed on the input(s). Let's take a look at an example. What is the difference between the following two functions?

• f(x) = x² + 3x
• f(y) = y² + 3y

The answer is that there is no difference between these two functions. In each case, we have a single input and have these instructions.

1. Square your input
2. Add three times your input

The fact that we use a different variable in the two cases is irrelevant. Each function has the same number of inputs, and we perform the same instructions on those inputs. The functions are thus equivalent. Let's look at another example.

• If f(x) = -x² + x + 17, then what is f(x - 2)?

Here are our instructions.

1. Take the negative of the square of your input
2. Add your input
3. Add 17

We are asked to find f(x - 2); our input has changed, but our instructions have not.

• f(x - 2) = -(x - 2)² + (x - 2) + 17
• f(x - 2) = -(x² - 4x + 4) + x - 2 + 17
• f(x - 2) = -x² + 4x - 4 + x + 15
• f(x - 2) = -x² + 5x + 11

Let's look at one more example. It's similar but a bit trickier.

• If f(x - 3) = x² + 3, then what is f(x)?

In this case, our input is x - 3, but we want to turn our input into just x. This may seem complicated, but it really is not. All we have to do is to substitute x + 3 everywhere that we see x. Let's do it.

• f((x + 3) - 3) = (x + 3)² + 3
• f(x + 3 - 3) = (x² + 6x + 9) + 3
• f(x) = x² + 6x + 12

Here are a couple of extra problems for you to try if you so please.

Example Problems

If f(x) = 2x² + 4x - 2, then what is f(x - 3)?

If f(x + 5) = x² + 14x + 53, then what is f(x)?