Understanding Functions In Algebra

In this article I discuss the history and uses of functions. Algebraic functions are some of the most important concepts in math. Since this is a series on Algebra, I use specific examples that are related.

Introduction

Functions are a type of relation that tie numbers together. They have to have a mutual meaning, but when they do, the data given can be very meaningful. Beginning functions often use tables for a visual aid. Tables in this fashion show the direct relationship between a set of numbers.

Notation and Pairs

When talking about functions, the notation y = f(x) is traditionally used. This means that when (x) is used as an input into (f), then the result is y. That is a fundamental relationship and must be understood. It is important to realize that a function returns a set of ordered pairs, like (x,y). A relation that has one output for each input is, by definition, a function. (x) is the independent variable and (y) is the dependent variable.

A set of inputs (x) is called the domain of the function. The set of related outputs (y) is the range of the function. A letter (f) is usually the name of a function. A function can be represented in words, a table, or a diagram. They all mean the same thing. Each method will have its limitations.

Functions can be represented verbally, graphically, symbolically, or numerically. There is no best way, it depends on the situation and the data you want to show.

Every function is a relation but not every relation is a function. This is important to remember. The domain is the set of all read numbers for which it is defined.

Constant Functions

I want to add some detail to our definition of functions. They can be considered in another light, that is, they can be linear or nonlinear. An example of a linear function is called a constant function. It should not be too hard to surmise what this is. A constant function is one who’s output remains constant. These are actually very common.

Linear Functions

This is a type of function where the output has the same rate of increasing or decreasing. It is represented by f(x) = ax + b. If a=0 at any point then we will just have a constant function. So in a linear function, each time x increases, f(x) changes by the amount of a.

Slope

The graph of a linear function is a straight line. Slope is the number that indicates the inclination of this line. If you have two points, slope = m = \frac{\delta y}{\delta x}. If this slope is positive, it rises from left to the right. However, if its negative, the line sinks from left to right.

Example 1

Find the slope of the line passing through the points (-2,3) and (1, -2).

You calculate it like this:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2-3}{1-(-2)} = \frac{-5}{3}$$

Slope like this is also known as rate of change of a function. It is also a constant function.

When using a graph to figure out estimates, the units will be easy and given for you. It will be the units for the y-axis over the x-axis. Take advantage of this.

Nonlinear Functions

Nonlinear functions are easy to explain. They are functions with variable output. So the output does not increase or decrease with any regular interval. If you were looking at a graph, the line would be curved. This is what a curved line means.

If you are looking at a table of data and wonder if it is linear or nonlinear, just examine the output. That part should be labeled clearly. The output will have the same rate of change if it is linear, otherwise it is nonlinear.

Average Rate of Change

Since the graphs of nonlinear functions are not straight, there is no single slope. Instead, there are many different slopes in a particular curve. However, we can take an average of the slopes to get a general idea. This is an important calculus topic too. If we have two points on a curve and draw a straight line connecting them, we call this a secant line. So the average rate of change is telling us, on average, how fast a quantity is changing. the average rate of change is just the slope of that particular line between two points.

When a function is constant, its average rate of change, or slope, is 0. In a linear function such as:

$$f(x) = ax + b$$

its average rate of change is equal to a, the slope of its graph. A nonlinear function has a variable rate of change.

Difference Quotient

The difference quotient is encountered a lot in calculus. It looks like this:

$$\frac{f(x + h) - f(x)}{h}$$

Example 2

Calculating A Difference Quotient

If $$f(x) = x^2 - 2x$$ Find f(x+h)

Our expression is $$x^2 - 2x$$ . We substitute (x+h) for every (x) in our expression.

$$f(x + h) = (x + h)^2 - 2(x + h)$$

Now multiple the expression out.

$$f(x + h) = x^2 + 2xh + h^2 -2x -2h$$

Ok that gives us $$f(x + h)$$ . Now we input that into our difference quotient.

$$\frac{ f(x + h) - f(x)}{h } = \frac{ x^2 + 2xh + h^2 -2x -2h- (x^2 - 2x) } {h}$$

Lets rewrite that ending part so everything has the correct sign.

$$\frac{ f(x + h) - f(x) } {h} = \frac{ x^2 + 2xh + h^2 -2x -2h -x^2 + 2x } {h}$$

It should be immediately obvious that a few things will cancel out. Lets do that.

$$\frac{ 2xh + h^2 -2h }{h}$$

$$\frac{ h(2x + h -2) } {h}$$

$$2x + h -2$$

This is our answer. These problems are usually designed so they cancel out certain values to make it work. So when working these, if the problem does not behave like that, you might have made a mistake and need to go back and look at it.

Example 3

The distance in feet that a racehorse travels is given by the function

$$d(t) = 2t^2$$

Find d(t + h).

Find the difference quotient.

We will be substituting (t+h) for t in the expression 2t ^2.

$$d(t+h) = 2(t+h)^2$$

$$2(t^2 +2th + h^2)$$

This gives us (2(t+h).

$$= 2t^2 + 4th + 2h^2$$

Now lets calculate the difference quotient.

$$\frac{d(t+h) - d(t)}{h} = \frac{2t^2 + 4th + 2h^2 -2t^2}{h}$$

Cancel stuff out and rewrite it.

$$\frac{4th + 2h^2}{h} = \frac{h(4t + 2h)}{h}$$

Cancel those h’s. We get:

$$4t + 2h$$

If t=7 and h=.01, then the difference quotient becomes:

$$4t + 2h = 4(7) + 2(.1) = 28.2$$

So functions are very interesting. They can tell us a nice picture on what each one means. Constant and linear functions are easy to understand but its also important to remember what they represent and how they look graphically.

The slope on a graph is the degree of the rise and fall of a line between two points. This is also the rate of change. Rate of change tells us how fast any quantity changes. This then leads us to difference quotients.