Hi. This math page will be about function notation.

**Table Of Contents**

**Introduction and Motivation**

In my experience, the function notation was introduced to me in one of my later years of high school mathematics. When and how this topic is taught varies between regions, school boards and countries.

When it comes to equations we usually use the variable to represent the dependent variable. There are times when there are multiple equations such as and . With those equations, when we substitute a value of 2 for for example we need to specify which equation this needs to be applied to.

Function notation makes it easier for the math reader/student to identify which equation is being used for the substitution. Instead of , we would use use and instead.

**The Function Notation Form**

Suppose we have a simple linear function of the form (instead of ). When we are substituting , we use the notation of . With substitutions in the function notation form, the independent variable is replaced by 3. This is much more efficient than writing down substituting into .

In , we have as the independent variable and as the dependent variable. We can also view the variable as an input into the function (machine) which gives a corresponding output

Suppose we have another function. We call this function . (You can use a different letter other than f.) Setting yields .

**Examples**

Here are a few more examples.

Suppose we have .

__Example One__

What is ?

__Example Two__

What is ?

We first evaluate the quantity inside the bracket of the function .

__Example Three__

What is h(a) where is some number?

Before, we substitute with a number. Here we do a similar procedure and substitute with .

__Example Four__

Suppose we want to double the quantity of our input. What is ?

Instead of , we have . Whenever there is an , replace it with .

**Practice Questions**

Suppose we have the functions and .

1) What is ?

2) Evaluate .

3) Evaluate .

4) Evaluate .

5) Evaluate . (Composition Functions: First evaluate and then .)

**Answers**

1) .

2) .

3) .

4) .

5) . Then