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Section1.4Gist of Functions: Definition and Notation

Subsection1.4.1Defining a Function

A function is a rule that may be in the form of a graph, or a table of values or a formula. It may even be a sentence or a set of instructions. A function takes an input value and uses the rule to create an output value.

Function notation looks like this: \begin{equation*} f(\text{input})=\text{output} \end{equation*}

But, instead of writing the words “input” and “output” we usually use variables, like \(x\) and \(y\text{.}\) Then we define in words what the variables actually mean or represent. Most often the notation will look something like this: \begin{equation*} f(x)=y \end{equation*}

Inside the parentheses is the independent variable, in this example it is \(x\text{.}\) Outside the parentheses, on the other side of the equals sign, is the dependent variable. In this case ours is \(y\text{.}\)

Subsection1.4.2Inputs have unique outputs

An important property of functions is that any single input value can only have one output value, and that output is unique.


When you buy a car it usually comes with a sticker on the window that includes information about the fuel efficiency. Cars are tested on a track under controlled conditions and most likely the manufacturer has all kinds of graphs and tables for the efficiency measures for various speeds of the car.

Let \(E\) be the fuel efficiency, in miles per gallon, of a car traveling at \(s\) miles per hour. Because knowing the speed of the vehicle determines its efficiency, we can use function notation to write \begin{equation*} E = f(s) \end{equation*} That is, efficiency is a function of speed.

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Figure1.4.4Graph of \(f(s)\text{:}\) Fuel Economy versus Speed

The input is the speed, \(s\text{,}\) and the output is the efficiency, \(E\) so we can write an ordered pair \((s, E)\) or \((s, f(s))\text{.}\)

It should make sense that a car can only have one efficiency at any given speed (one input, one output) since it cannot get 10 mpg and 20 mpg at the same time.

Also, during the controlled test, if the car traveling at a speed of 5 mph has an efficiency of 12 mpg then 5 mph always gets us 12 mpg. The value \(12\) is the unique output associated with the input of \(5\text{.}\)

However, it is possible there are multiple inputs that give you the same output. From the graph we see that a fuel efficiency of 25 mpg can be attained by traveling at a speed of about 16 mph and also at about 70 mph.

Graphically, a relation must pass the vertical line test 1.1.13 to be a function. When the test is “passed”, we say the function is well defined.

Subsection1.4.3Describing functions with intervals

A function may have many different characteristics:

  • Output values may increase or decrease.

  • The graph may curve up or curve down.

  • The outputs may be positive or negative.

When describing these characteristics for a function, we typically refer to an interval (connected section) of the input axis, or to a particular value of the input where something interesting occurs. Figure 1.4.4

We can use inequality symbols to describe an interval where a function has a particular property, like increasing 1.2.3 for \(5\lt s\lt55\text{.}\) This means the function values are getting bigger as the inputs change from \(5\) to \(55\text{.}\)

Another characteristic we can describe with inequalities is concavity. We say a function is concave up if its graph curves upward from left to right, and that it is concave down if it curves downward from left to right.


The efficiency graph has varying concavity 1.2.4 depending on what section of inputs we use:

  • concave down for \(5\lt s\lt60\text{.}\)

  • concave up for \(60\lt s\lt75\text{.}\)

Subsection1.4.4Evaluate versus Solve

The word evaluate means to use a known input value to find the output value. Evaluating a function looks like this: \begin{equation*} f(35) \end{equation*} It means, “Find the result of choosing 35 mph” as the input. The expression \(f(35)\) represents the output, which we can see from the graph.

Solve means the output is already known and we are trying to find all the possible inputs that give us that output. These input values are called solutions of the equation. For instance, we may be asked to “Solve \(f(s)=35\text{.}\)”

This means that we are to find all input values (solutions) that would give us the output \(35\text{.}\)

Using the graph of the efficiency function we see that the equation \(f(s)=35\) has two solutions. There are two speeds at which the car has an efficiency of 35 mpg.

In fact, solving graphically is often a preferred method of solving equations because the output is essentially the height of the graph. So when we solve \(f(s) = 35\) graphically, we are asking “What values of \(s\) make the graph \(35\) units high?”

Subsection1.4.5Average Rate of Change

Average rate of change is very important topic of this course. You will see it over and over again in various forms. Basically it means slope, but it's more than that. It's also a notation and a concept that sets the stage for calculus.

Using our vehicle efficiency function we know the car's efficiency at various speeds. A change in speed may or may not have much affect on the efficiency. The average rate of change of the efficiency is actually the measure of slope between any two speeds.

The calculation is always a change in output divided by a change in input.

Gathering information from the function we can calculate changes in efficiency due to changes in speed.

Using function notation we write \begin{align*} \frac{\Delta f}{\Delta s}\amp=\frac{f(10)-f(20)}{10-20}\\ \amp=1.1 \end{align*} The number \(1.1\) (notice it is positive) means that between speeds of 10 mph and 20 mph, the car's efficiency is increasing at an average rate of 1.1 mpg per 1 mph.

On a different interval we could write \begin{align*} \frac{\Delta f}{\Delta s}\amp=\frac{f(60)-f(72)}{60-72}\\ \amp=-0.8 \end{align*} The number \(-0.8\) (notice it is negative) means that between speeds of 60 mph and 72 mph, the car's efficiency is decreasing at an average rate of -0.8 mpg per 1 mph.

The little triangle symbol \(\Delta\) is also very important. It means, “change in”. Therefore based on our example, \(\Delta f\) means “change in efficiency” and \(\Delta s\) means “change in speed”. Together the symbol \(\frac{\Delta f}{\Delta s}\) is called “average rate of change”.


Units are the “names” of the quantities we are counting. They give meaning to the numbers or values we use in an expression.

When you measure the width of a room, the tape measure might count 10.5 ft. The number \(10.5\) is the value and “ft” is the unit.

In car efficiency example the units of efficiency are “mpg” (miles per gallon) and the units of speed are “mph” (miles per hour). Therefore the average rate of change for this example will have units of mpgmph.

Because slope is just “rise over run” — a fraction, the units of the average rate of change are also a fraction.


Evaluating and Making Combinations

Evaluating Functions and Solving for an Unknown

Additional Problems