One-to-many rules, many-to-one and one-to-one functions

1 One-to-many rules, many-to-one and one-to-one functions

One-to-many rules

Recall from Section 2.1 that a rule for a function must produce a single output for a given input. Not all rules satisfy this criterion. For example, the rule ‘take the square root of the input’ cannot be a rule for a function because for a given input there are two outputs; an input of 4 produces outputs of 2 and $- 2$ . Figure 10 shows two ways in which we can picture this situation, the first being a block diagram, and the second using two sets representing input and output values and the relationship between them.

Figure 10 :

{ This rule cannot be a function - it is a one-to-many rule}

Such a rule is described as a one-to-many rule . This means that one input produces more than one output. This is obvious from inspecting the sets in Figure 10.

The graph of the rule ‘take $\pm \sqrt{x}$ ’ can be drawn by constructing a table of values:

Table 4

$x$	0	1	2	3	4
$y = \pm \sqrt{x}$	0	$\pm 1$	$\pm \sqrt{2}$	$\pm \sqrt{3}$	$\pm 2$

The graph is shown in Figure 11(a). For each value of $x$ there are two corresponding values of $y$ . Plotting a graph of a one-to-many rule will result in a curve through which a vertical line can be drawn which cuts the curve more than once as you can see. The vertical line cuts the curve more than once because there is more than one $y$ value for each $x$ value.

Figure 11

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By describing a rule more carefully it is possible to make sure a single output results from a single input, thereby defining a valid rule for a function. For example, the rule ‘take the positive square root of the input’ is a valid function rule because a given input produces a single output. The graph of this function is displayed in Figure 11(b).

Many-to-one and one-to-one functions

Consider the function $y (x) = x^{2}$ . An input of $x = 3$ produces an output of 9. Similarly, an input of $- 3$ also produces an output of 9. In general, a function for which different inputs can produce the same output is called a many-to-one function . This is represented pictorially in Figure 12 from which it is clear why we call this a many-to-one function.

Figure 12 :

{ This represents a many-to-one function}

Note that whilst this is many-to-one it is still a function since any chosen input value has only one arrow emerging from it. Thus there is a single output for each input.

It is possible to decide if a function is many-to-one by examining its graph. Consider the graph of $y = x^{2}$ shown in Figure 13.

Figure 13 :

${ The function $y=x^2$ is a many-to-one function}$

We see that a horizontal line drawn on the graph cuts it more than once. This means that two (or more) different inputs have yielded the same output and so the function is many-to-one.

If a function is not many-to-one then it is said to be one-to-one . This means that each different input to the function yields a different output.

Consider the function $y (x) = x^{3}$ which is shown in Figure 14. A horizontal line drawn on this graph will intersect the curve only once. This means that each input value of $x$ yields a different output value for $y$ .

Figure 14 :

${ The function $y(x)=x^3$ is a one-to-one function}$

Task!

Study the graphs shown in Figure 15. Decide which, if any, are graphs of functions. For those which are, state if the function is one-to-one or many-to-one.

Figure 15

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not a function,
one-to-one function,
many-to-one function

1 One-to-many rules, many-to-one and one-to-one functions

Task!

Answer