How do tables and graphs represent relations

A General Note: Function A function is a relation in which each possible input value leads to exactly one output value. How To: Given a relationship between two quantities, determine whether the relationship is a function. Identify the input values. Identify the output values. If each input value leads to only one output value, classify the relationship as a function.

If any input value leads to two or more outputs, do not classify the relationship as a function. Is price a function of the item?

Is the item a function of the price? Figure 2. The output values are then the prices. See Figure 2. Figure 3. Percent Grade 0—56 57—61 62—66 67—71 72—77 78—86 87—91 92— Grade Point Average 0. Solution For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade.

Try It 1 The table below lists the five greatest baseball players of all time in order of rank. Using Function Notation Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers.

Example 3: Using Function Notation for Days in a Month Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Figure 4. Analysis of the Solution Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output.

How To: Given a table of input and output values, determine whether the table represents a function. Identify the input and output values. Check to see if each input value is paired with only one output value. If so, the table represents a function. Example 5: Identifying Tables that Represent Functions Which table, a , b , or c , represents a function if any? Is the area of a circle a function of its radius?

If yes, is the function one-to-one? If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Yes, letter grade is a function of percent grade; b. No, it is not one-to-one. There are different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade. As we have seen in some examples above, we can represent a function using a graph.

Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.

Howto: Given a graph, use the vertical line test to determine if the graph represents a function. If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. From this we can conclude that these two graphs represent functions.

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Howto: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. Are either of the functions one-to-one? Any horizontal line will intersect a diagonal line at most once. In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers.

When working with functions, it is similarly helpful to have a base set of building-block elements. Some of these functions are programmed to individual buttons on many calculators. We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book.

It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. Jay Abramson Arizona State University with contributing authors. Learning Objectives Determine whether a relation represents a function. Find the value of a function. Determine whether a function is one-to-one. Use the vertical line test to identify functions. Graph the functions listed in the library of functions.

In this case, each input is associated with a single output. Function A function is a relation in which each possible input value leads to exactly one output value. How To: Given a relationship between two quantities, determine whether the relationship is a function Identify the input values. Identify the output values.

If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function. Is price a function of the item? Is the item a function of the price? The output values are then the prices. Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it.

Therefore, the item is a not a function of price. Percent grade 0—56 57—61 62—66 67—71 72—77 78—86 87—91 92— Grade point average 0. Is the player name a function of the rank? Answer a Yes Answer b yes. Using Function Notation Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers.

Solution Using Function Notation for Days in a Month Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Analysis Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. Representing Functions Using Tables A common method of representing functions is in the form of a table.

Check to see if each input value is paired with only one output value. If so, the table represents a function. Finding Input and Output Values of a Function When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluation of Functions in Algebraic Forms When we have a function in formula form, it is usually a simple matter to evaluate the function.

How To: Given the formula for a function, evaluate. Given the formula for a function, evaluate. In this course, you will see brackets used to indicate a set of ordered pairs for a relation, as well as sets of numbers.

Two ways that we can represent a relation that are more organized than a list of ordered pairs are with tables and arrow diagrams. The table and arrow diagram for this relation are shown below. Arrow diagrams are less common than the other representations in this section, but are particularly useful when talking about functions, which will be discussed in Chapter 4.

The last and perhaps most important representation of a relation that we will discuss is a graph. For relations between sets of numers, graphs are a visual way to represent the relationship between the numbers on a coordinate plane. The rectangular coordinate system consists of two real number lines that intersect at a right angle. These two number lines define a flat surface called a plane.