Review of Functions: Learn It 1

  • Determine the set of all possible inputs (domain) and outputs (range) for a function from its graph or equation
  • Find where functions cross the x-axis and y-axis by looking at equations, graphs, and data tables
  • Interpret graphs and tables to describe function behaviors, including symmetry
  • Combine two or more functions to create a new function

Functions

Given two sets [latex]A[/latex] and [latex]B[/latex], a set with elements that are ordered pairs [latex](x,y)[/latex], where [latex]x[/latex] is an element of [latex]A[/latex] and [latex]y[/latex] is an element of [latex]B[/latex], is a relation from [latex]A[/latex] to [latex]B[/latex]. A relation from [latex]A[/latex] to [latex]B[/latex] defines a relationship between those two sets.

A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input.

The area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

functions

A function [latex]f[/latex] consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output.

For a general function [latex]f[/latex] with domain [latex]D[/latex], we often use [latex]x[/latex] to denote the input and [latex]y[/latex] to denote the output associated with [latex]x[/latex]. When doing so, we refer to [latex]x[/latex] as the independent variable and [latex]y[/latex] as the dependent variable, because it depends on [latex]x[/latex]. Using function notation, we write [latex]y=f(x)[/latex], and we read this equation as “[latex]y[/latex] equals [latex]f[/latex] of [latex]x[/latex].” 

The concept of a function can be visualized using Figure 1.

An image with three items. The first item is text that reads “Input, x”. An arrow points from the first item to the second item, which is a box with the label “function”. An arrow points from the second item to the third item, which is text that reads “Output, f(x)”.
Figure 1. A function can be visualized as an input/output device.

Evaluating a Function

Evaluating a function is like finding out what the function does when you give it a specific input. Think of a function as a machine in a factory: you put something in, the machine works on it, and then it gives you something back. In the case of a function, you give it a number, and it gives you another number according to a specific rule.

How to: Evaluate a Function:

  1. Identify the input: This is the value that you will put into the function, often represented as ‘[latex]x[/latex]‘.
  2. Plug the input into the function: Replace the ‘[latex]x[/latex]‘ in the function’s formula with the value of your input.
  3. Follow the operations: Perform the mathematical operations in the formula with your input value. This means you’ll do any addition, subtraction, multiplication, division, exponentiation, etc., that the function tells you to do with that input.
  4. Simplify: If the function’s rule has more than one operation, follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction) to simplify the expression down to a single number.
  5. Find the output: The number you end up with after doing all the operations is the output of the function, often represented as ‘[latex]f(x)[/latex]‘ or ‘[latex]y[/latex]‘.
For the function [latex]f(x)=3x^2+2x-1[/latex], evaluate,

  1. [latex]f(-2)[/latex]
  2. [latex]f(\sqrt{2})[/latex]
  3. [latex]f(a+h)[/latex]

Representing Functions

Typically, a function is represented using one or more of the following tools:

  • A table
  • A graph
  • A formula

We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.

Tables

Functions described using a table of values arise frequently in real-world applications. Consider the following simple example.

We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable [latex]x[/latex] be the time after midnight, measured in hours, and the output variable [latex]y[/latex] be the temperature [latex]x[/latex] hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.

Table 1. Temperature as a Function of Time of Day
Hours after Midnight Temperature [latex](\text{°}F)[/latex] Hours after Midnight Temperature [latex](\text{°}F)[/latex]
[latex]0[/latex] [latex]58[/latex] [latex]12[/latex] [latex]84[/latex]
[latex]1[/latex] [latex]54[/latex] [latex]13[/latex] [latex]85[/latex]
[latex]2[/latex] [latex]53[/latex] [latex]14[/latex] [latex]85[/latex]
[latex]3[/latex] [latex]52[/latex] [latex]15[/latex] [latex]83[/latex]
[latex]4[/latex] [latex]52[/latex] [latex]16[/latex] [latex]82[/latex]
[latex]5[/latex] [latex]55[/latex] [latex]17[/latex] [latex]80[/latex]
[latex]6[/latex] [latex]60[/latex] [latex]18[/latex] [latex]77[/latex]
[latex]7[/latex] [latex]64[/latex] [latex]19[/latex] [latex]74[/latex]
[latex]8[/latex] [latex]72[/latex] [latex]20[/latex] [latex]69[/latex]
[latex]9[/latex] [latex]75[/latex] [latex]21[/latex] [latex]65[/latex]
[latex]10[/latex] [latex]78[/latex] [latex]22[/latex] [latex]60[/latex]
[latex]11[/latex] [latex]80[/latex] [latex]23[/latex] [latex]58[/latex]

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.


Graphs

Given a function [latex]f[/latex] described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table 1 can give us a better idea of their fluctuation throughout the day. Figure 5 shows the plot of the temperature function.

An image of a graph. The y axis runs from 0 to 90 and has the label “Temperature in Fahrenheit”. The x axis runs from 0 to 24 and has the label “hours after midnight”. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58).
Figure 5. The graph of the data from Table 1 shows temperature as a function of time.
 

From the points plotted on the graph in Figure 5, we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. 

An image of a graph. The y axis runs from 0 to 90 and has the label “Temperature in Fahrenheit”. The x axis runs from 0 to 24 and has the label “hours after midnight”. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58). A line connects all the points on the graph.
Figure 6. Connecting the dots in Figure 5 shows the general pattern of the data.

In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern.

Algebraic Formulas

Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. 

The area of a circle of radius [latex]r[/latex] is given by the formula [latex]A(r)=\pi r^2[/latex]. When an object is thrown upward from the ground with an initial velocity [latex]v_{0}[/latex] ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula [latex]s(t)=-16t^2+v_{0}t[/latex]. When [latex]P[/latex] dollars are invested in an account at an annual interest rate [latex]r[/latex] compounded continuously, the amount of money after [latex]t[/latex] years is given by the formula [latex]A(t)=Pe^{rt}[/latex]. Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

Given an algebraic formula for a function [latex]f[/latex], the graph of [latex]f[/latex] is the set of points [latex](x,f(x))[/latex], where [latex]x[/latex] is in the domain of [latex]f[/latex] and [latex]f(x)[/latex] is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of [latex]f[/latex] consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.