# How to Identify Families of Functions

Instructor: David Karsner
Functions come in all kinds of varieties, but they can be grouped together into families that have common characteristics. The difference and similarities can be seen in the equations and graphs of these functions. In this lesson, you will learn about the characteristics of different families of functions and how to recognize them based on their equations and graphs.

## Lions, Tigers, and Bears, Oh My!

Most likely, at a very early age. you would go and visit the zoo and noticed that all animals were not the same. Some animals were just a little different and some animals were a lot different. Lions and tigers are more like each other than they are a bear. Lions, tigers, and bears are more like each other than, say a jellyfish. We have grouped the animals into categories, based on their characteristics. We can group functions together in a very similar way. Functions can be grouped together into families, based on their characteristics and the appearance of their graphs. This lesson will give you the clues to look for to identify which family the function belongs to.

## What is a Function?

A function is a relationship between one or more variables. A function maps one input value to exactly one output value. The common notation for a function is f(x); however, you will see other letters used, such as g(x), h(x) and many others. An example of a function would be f(x)=4x-2, x is the input value and f(x) is the output value. The notation f(x) is also referred to as the y value, the dependent variable.

## What is a Family of Functions?

There are many different kinds of functions. These different types of functions can be grouped together into several different categories. The categories are called families of functions. Many of the categories are based on the degree of the function (the largest exponent of the function).

## Polynomials

Polynomials are a large family of functions. It is an umbrella covering several sub-categories. A polynomial is a function that has one or more terms connected by addition or subtraction. Every term must have an exponent that is a positive whole number. The domain of all polynomials are the real numbers (all values of x can be used). The graph of a polynomials is continuous. The functions f(x)=4x-2 and g(x)=2x3-9x-8 are polynomials. h(x)= 4x-2+2x +4 is not a polynomial. The easiest way to identify the different sub-categories of polynomials is to look at the largest exponent in the function.

### Linear

The first sub-category of polynomials is the linear functions. Linear functions have a degree of one (the exponent on the variable is usually not written). Linear functions can be written in the form f(x)=mx+b, where m is the slope of the function and b is the y-intercept. The graph of a linear function is a line. Examples of linear functions are f(x)=-3x+7, and m(x)=10-5x.

A quadratic function is a polynomial that has a degree of two. The standard form of a quadratic is f(x)=ax2+bx+c. The values of a,b,and c allow you to determine the vertex, zeros, concavity, and y-intercepts of the quadratic function. The graph of a quadratic is a parabola (similiar to the letter U). Examples of quadratic functions are f(x)=3x2+7x+10 and j(x)=-7x2+100.

### Cubic

A cubic function is a polynomial that has a degree of three. To identify a cubic function, check to see if the largest exponent is three and it is polynomial. The graph of a cubic function looks similar to a sideway z or a chair. Examples of cubic functions are f(x)=5x3+2x2-4x+10 and g(x)=7x3-9x2+x-2.

## Rational

The easiest way to identify a rational function is that they have a variable in the denominator. If there is only a constant in the denominator, it is not a rational function. Unlike the polynomials, the domain of a rational function is NOT all real numbers. Most likely the denominator at some value of x will equal zero. At this point, the function is undefined. The graph of these functions will have gaps (it will not be continuous) at the x value that causes the denominator to be equal to zero. These gaps are called vertical asymptotes.

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