Lesson Video: Identifying Functions Mathematics

In this video, we will learn how to identify a function from a relation shown by a set of ordered pairs, a relation diagram, an equation, or a graph.

15:51

Video Transcript

In this video, we will learn how to identify a function from a relation shown by a set of ordered pairs, a relation diagram, an equation, or a graph.

We recall that a relation or mapping transforms elements from one set onto elements from another. If every input in this mapping has exactly one output, it is called a function. Let’s begin by defining this formally. A function maps every element from an input set onto exactly one element from an output set. Functions can either be one to one, where one input has one output, or many to one, where many inputs map onto the same output. If a function 𝑓 maps elements of π‘₯ to 𝑦, we can use the following notation. Functions can be represented by ordered pairs of numbers, mapping diagrams, equations, and graphs. In our first example, we will demonstrate how to apply this definition of a function to determine whether a set of ordered pairs defines a function.

Which of the following relations represents a function?

In this question, we are given two relations 𝐴 and 𝐡, both of which consist of five ordered pairs. We begin by recalling that a function is a rule that takes every member of a set and maps it onto exactly one member in a second set. For a group of ordered pairs π‘₯, 𝑦, the input is π‘₯ and the output is 𝑦. This means that for a group of ordered pairs to represent a function, no two ordered pairs can have the same input with a different output. In other words, if an ordered pair shares the same π‘₯-value, their 𝑦-value must also be the same.

We observe that relation 𝐴 has two ordered pairs with an π‘₯-value of four. For relation 𝐴 to be a function, the corresponding 𝑦-values in these ordered pairs must also be the same. However, the ordered pairs are four, 12 and four, 15. Since the 𝑦-values are not the same, relation 𝐴 cannot represent a function. Relation 𝐴 also has two ordered pairs with an π‘₯-value of five. They too have different 𝑦-values: the ordered pairs five, 18 and five, 21. We can therefore conclude that relation 𝐴 does not represent a function.

The ordered pairs of relation 𝐡 have unique π‘₯-values: the integers four, five, six, seven, and eight. This satisfies the condition that a function takes every member of a set and maps it onto exactly one member in the second set. We can therefore conclude that the relation that represents a function is relation 𝐡.

We could also represent these ordered pairs on a mapping diagram. If we begin by considering relation 𝐴, we observe that elements four and five from the input or π‘₯-column map to more than one element in the 𝑦- or output column. This mapping diagram therefore cannot represent a function. For a mapping diagram to represent a function, every input value must have a single output. When we consider relation 𝐡, we see that every input, the integers four, five, six, seven, and eight, have a single output, the integers 12, 15, 18, 21, and 24. Whilst it is not required for this question, we notice that the 𝑦-values are three times the π‘₯-values. The function can therefore be written as the equation 𝑦 is equal to three π‘₯.

In this first example, we have seen that we can identify whether a relation is a function using ordered pairs or a mapping diagram. Let’s now consider what happens when a relation is represented by a graph. Since a function can be represented by a set of ordered pairs, it follows that we can also use a graph as a visual representation of a function. The graph of a function 𝑓 is defined by a set of ordered pairs π‘₯, 𝑦 such that 𝑦 equals 𝑓 of π‘₯. For example, consider the function 𝑓 that maps π‘₯ onto π‘₯ squared. The set of ordered pairs that represents this function is given by π‘₯, π‘₯ squared. And the graph of the function is given by the graph of 𝑦 equals π‘₯ squared as shown.

This is an example of a many-to-one function as there are values in the range of the function that are associated with more than one value in the domain. For example, the points two, four and negative two, four have the same 𝑦-value but a different π‘₯-value. To establish whether a graph represents a function or not, we use the vertical line test. In order to carry out the vertical line test, we begin by drawing a line parallel to the 𝑦-axis. If the vertical line intersects the curve more than once, the graph does not represent a function. However, if the line does intersect the curve exactly once for a given set of inputs, then the graph does represent a function.

Let’s consider two examples. In the first figure, a vertical line intersects the curve twice. This means that this graph does not represent a function. However, in the second figure, any vertical line will intersect the curve exactly once. This means that this graph does represent a function. We will now look at an application of the vertical line test.

Which of the following relations represents a function given that π‘₯ is the input and 𝑦 is the output?

To answer this question, we will use the vertical line test. This is a graphical way of working out whether a graph represents a function. If a graph represents a function, any vertical line will intersect the function at most once. We will therefore add a vertical line to our diagrams and inspect the number of points of intersection with the curves. In figure (A), we observe that if we add the vertical line π‘₯ equals five, the line intersects the curve twice. This means that graph (A) cannot represent a function. The vertical line π‘₯ equals negative four intersects the curve of graph (B) exactly once. We can translate this vertical line by any distance horizontally, and this will always be the case. We can therefore conclude that graph (B) represents a function. The correct answer is (B).

Our definition of a function says that it is some rule that maps an input onto exactly one output. In many cases, we can define that rule algebraically. For example, graph (B) represents a function of the graph 𝑦 equals two π‘₯ plus four. Let’s clear some space and see how we can define this as a function. We can define the function using functional notation as 𝑓 of π‘₯ is equal to two π‘₯ plus four. Alternatively, we can use arrow notation. This means that we can determine whether 𝑓 represents a function by plotting the graph of 𝑦 equals 𝑓 of π‘₯ and performing the vertical line test.

Can the equation π‘₯ squared plus 𝑦 squared equals four be expressed as a function? If yes, state the function.

We recall that the vertical line test can be used to determine whether a graph represents a function. This means that in order to establish whether the equation π‘₯ squared plus 𝑦 squared equals four can be expressed as a function, we can sketch its graph. Any equation in the form π‘₯ squared plus 𝑦 squared equals π‘Ÿ squared represents a circle which is centered at the origin with radius π‘Ÿ. In this question, as π‘Ÿ squared is equal to four and π‘Ÿ must be positive, the radius of our circle is two. Sketching the circle, we see that it intersects the π‘₯- and 𝑦-axis at two and negative two.

Adding a vertical line to our diagram, we note that it intersects the circle twice. As there is more than one point of intersection, the graph does not represent a function. We can therefore conclude that the equation π‘₯ squared plus 𝑦 squared equals four cannot be expressed as a function. The correct answer is no.

In this question, it was fairly easy to sketch the graph of the equation so we could perform the vertical line test. However, we might have instead attempted to rewrite the equation in terms of π‘₯. Subtracting π‘₯ squared from both sides of our equation gives us 𝑦 squared is equal to four minus π‘₯ squared. Square rooting both sides of this equation gives us 𝑦 is equal to positive or negative square root of four minus π‘₯ squared. We notice that when we take the square root of both sides of the equation, we take the positive and negative square root of the expression four minus π‘₯ squared. This means that for any single value of π‘₯, there could be two possible outputs, a positive and a negative. Since a function maps every element from one set onto exactly one element from a second set, we can deduce that the equation π‘₯ squared plus 𝑦 squared equals four cannot represent a function.

In our final example, we will demonstrate how to perform a similar process to represent an equation in function notation.

Which of the following is the equation 𝑦 cubed equals π‘₯ squared plus one expressed in function notation? Is it (A) 𝑓 of π‘₯ equals π‘₯ squared plus one? (B) 𝑓 of π‘₯ equals π‘₯ squared plus one all cubed. Is it (C) this cannot be expressed as a function? (D) 𝑓 of π‘₯ equals the cube root of π‘₯ squared plus one. Or (E) 𝑓 of 𝑦 is equal to π‘₯ squared plus one.

To write any equation in function notation, we need to see whether we can express 𝑦 as a function of π‘₯ such that 𝑦 is equal to 𝑓 of π‘₯. We will therefore rearrange the equation to make 𝑦 the subject. In order to do this, we will cube root both sides of the equation. This gives us 𝑦 is equal to the cube root of π‘₯ squared plus one. We observe that 𝑦 is now written as a function of π‘₯. In other words, to find the value of 𝑦, we can substitute a value of π‘₯ into the expression the cube root of π‘₯ squared plus one. To express 𝑦 cubed equals π‘₯ squared plus one in function notation, we have 𝑓 of π‘₯ is equal to the cube root of π‘₯ squared plus one. The correct answer is option (D).

It is important to note that rearranging an equation to write 𝑦 as a function of π‘₯ will not always result in a function. We must ensure that any π‘₯-value from the domain of that function gives us a unique 𝑦-value when substituted in. For this reason, any rearrangement that results in needing to invert an even exponent, for example, the square root or fourth root, will not result in a function.

We will now finish this video by recapping some key points. A function maps every element from an input set onto exactly one element from an output set. Functions can either be one to one, one input has one output, or many to one, where many inputs map onto the same output. We have seen that functions can be represented by ordered pairs of numbers, mapping diagrams, equations, and graphs. The function notation 𝑦 equals 𝑓 of π‘₯ means that 𝑦 is a function of π‘₯. The arrow notation shown means that the function 𝑓 maps elements of set π‘₯ onto an element of set 𝑦. We saw in this video that there were several ways to test whether a relation is a function. One way to do this graphically was using the vertical line test.

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