Which relation is displayed in the graph

The domain is defined as all the possible input values usually x which allow the formula to work. Note that values that cause a denominator to be zero, which makes the function undefined, are not allowable values.

The range is the set of all possible output values usually y , which result from using the formula. The range of this function is all real numbers from -2 onward. We can express this using the interval. This is a function.

For any value of x you plug in, you will get only one possible value for y. This is not a function. Any value of x can give you more than one possible y. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. How To: Given a table of input and output values, determine whether the table represents a function. In both, each input value corresponds to exactly one output value.

When a table represents a function, corresponding input and output values can also be specified using function notation. When we know an input value and want to determine the corresponding output value for a function, we evaluate the function.

Evaluating will always produce one result because each input value of a function corresponds to exactly one output value. Solving can produce more than one solution because different input values can produce the same output value. When we have a function in formula form, it is usually a simple matter to evaluate the function. Because the input value is a number, 2, we can use simple algebra to simplify. In this case, the input value is a letter so we cannot simplify the answer any further.

In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that. This gives us two solutions. Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form.

How to: Given a function in equation form, write its algebraic formula. It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula. Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula? Yes, this can happen.

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them?

There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. This is meager compared to a cat, whose memory span lasts for 16 hours. At times, evaluating a function in table form may be more useful than using equations. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table.

The tabular form for function P seems ideally suited to this function, more so than writing it in paragraph or function form. How To: Given a function represented by a table, identify specific output and input values. Find the given input in the row or column of input values.

Identify the corresponding output value paired with that input value. Find the given output values in the row or column of output values, noting every time that output value appears. Identify the input value s corresponding to the given output value. Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph.

Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value s.

Some functions have a given output value that corresponds to two or more input values. However, some functions have only one input value for each output value, as well as having only one output for each input.

We call these functions one-to-one functions. This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

The function in part b shows a relationship that is a one-to-one function because each input is associated with a single output. A one-to-one function is a function in which each output value corresponds to exactly one input value. 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?