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Functions & Calculus Notations

As you move into quantitative subjects like calculus and probability, the notation becomes slightly more complex, but it still relies on a few fundamental building blocks.

Function Notation: f(x)

The notation (f(x)) is universally used to describe a function. It is read as "f of x". Think of a function as a machine: you put an input in ((x)), the machine does some work on it ((f)), and spits out a result ((y)). So, (f(x) = 2x + 1) means "whatever number you put in for (x), multiply it by 2 and add 1". If (x) is 3, then (f(3) = 7).

Summation: Sigma ((\Sigma))

When you need to add up a long list of numbers, mathematicians use the uppercase Greek letter Sigma ((\Sigma)). It simply means "Sum". A typical summation looks like this: [\sum_^ x_i] This looks terrifying, but it just means: "Start at (i=1), go up to (n), and add all the (x)'s together". So it equals (x_1 + x_2 + x_3 + \dots + x_n). You'll see this everywhere in statistics when calculating averages.

Product: Pi ((\Pi))

Similar to Sigma, the uppercase Greek letter Pi (Π) is used when you need to Multiply a long sequence of numbers together. [\prod_^ x_i] This means: (x_1 \times x_2 \times x_3 \times \dots \times x_n). This is frequently used when calculating compound returns over multiple time periods.

Limits and Infinity ((\infty))

Calculus often deals with what happens when numbers get incredibly large or incredibly small.

  • Infinity ((\infty)): This symbol looks like a sideways 8. It does not represent a specific number, but rather the concept of an endless, limitless quantity.
  • Limits ((\lim)): A limit asks the question, "What value is this function approaching as (x) gets closer to a certain point?" You might see: (\lim_{x \to \infty} f(x)). This is read as "The limit of f of x, as x approaches infinity." In plain English, it means "What happens to the result of this function as our input (x) gets bigger and bigger forever?"

Mini-Quiz

Question 1

How is the mathematical expression (f(x)) read aloud?

  1. f times x
  2. f of x
  3. function to the power of x
  4. the derivative of x

Correct Answer: B Explanation: (f(x)) is standard function notation and is universally read as "f of x".

Question 2

What mathematical operation is represented by the uppercase Greek letter Sigma ((\Sigma))?

  1. Multiplication of a series of numbers
  2. Division of a series of numbers
  3. Addition (summation) of a series of numbers
  4. Taking the square root of a number

Correct Answer: C Explanation: The capital (\Sigma) stands for Summation, meaning you add together a sequence of values.

Question 3

In a formula, what does the uppercase Greek letter Pi (Π) instruct you to do with a sequence of numbers?

  1. Add them together
  2. Multiply them together
  3. Find their average
  4. Subtract the smallest from the largest

Correct Answer: B Explanation: While (\Sigma) is used for summing numbers, the capital Π is used to indicate the product, meaning you multiply the sequence of numbers together.