Hi. This math page will be about function notation.

 

Table Of Contents

 

 

Introduction and Motivation

 

In my experience, the function notation \(f(x)\) 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 \(y\) to represent the dependent variable. There are times when there are multiple equations such as \(y = 2x + 5\) and \(y = x^2 + 4x + 3\). With those equations, when we substitute a value of 2 for \(x\) for example we need to specify which equation this \(x = 2\) 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 \(y\), we would use use \(f(x) = 2x +5\) and \(g(x) = x^2 + 4x + 3\) instead.

 

The Function Notation Form

 

Suppose we have a simple linear function of the form \(f(x) = x + 2\) (instead of \(y = x + 2\)). When we are substituting \(x = 3\), we use the notation of \(f(3) = 3 + 2 = 5\). With substitutions in the function notation form, the independent variable \(x\) is replaced by 3. This is much more efficient than writing down substituting \(x = 3\) into \(y = x + 2\).

In \(f(x) = x + 2\), we have \(x\) as the independent variable and \(y = f(x)\) as the dependent variable. We can also view the variable \(x\) as an input into the function (machine) \(f\) which gives a corresponding output \(y = f(x)\)

Imae Source

Source: https://s-media-cache-ak0.pinimg.com/originals/54/86/13/54861315402fa1605a5d570b2d1770d0.jpg

Suppose we have another function. We call this function \(g(x) = x^2 + 5\). (You can use a different letter other than f.) Setting \(x = 5\) yields \(g(5) = 5^2 + 5 = 25 + 5 = 30\).

 

Examples

 

Here are a few more examples.

Suppose we have \(h(x) = x^2 + 2x - 3\).

Example One

 

What is \(h(-2)\)?

 

\[h(-2) = (-2)^2 + 2(-2) - 3 = 4 - 4 - 3 = -3\]  

Example Two

 

What is \(h(1 + 3 * 2)\)?

We first evaluate the quantity inside the bracket of the function \(h\).

 

\[h(1 + 3 * 2) = h(1 + 6) = h(7) = 7^2 + 2(7) - 3 = 49 + 14 - 3 = 60\]

 

Example Three

What is h(a) where \(a\) is some number?

Before, we substitute \(x\) with a number. Here we do a similar procedure and substitute \(x\) with \(a\).

 

\[h(a) = a^2 + 2a - 3\]

 

Example Four

Suppose we want to double the quantity of our input. What is \(h(2x)\)?

Instead of \(x\), we have \(2x\). Whenever there is an \(x\), replace it with \(2x\).

 

\[h(2x) = (2x)^2 + 2(2x) - 3 = 4x^2 + 4x - 3\]

 

Practice Questions

 

Suppose we have the functions \(f(x) = x + 7\) and \(g(x) = x^2 - 3\).

  1. What is \(f(-4)\)?

  2. Evaluate \(g(1)\).

  3. Evaluate \(g(2^3 + 1)\).

  4. Evaluate \(f(x + h)\).

  5. Evaluate \(g(f(1))\). (Composition Functions: First evaluate \(f(1)\) and then \(g(f(1))\).)

 

Answers

 

  1. \(f(-4) = 3\).

  2. \(g(1) = -2\).

  3. \(g(2^3 + 1) = g(8 + 1) = g(9) = 81 - 3 = 78\).

  4. \(f(x + h) = x + h + 7\).

  5. \(f(1) = 8\). Then \(g(f(1)) = g(8) = 64 - 3 = 61.\)