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. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables [latex]h[/latex] for height and [latex]a[/latex] for age. The letters [latex]f,\text{ }g,[/latex] and [latex]h[/latex] are often used to represent functions just as we use [latex]x,\text{ }y,[/latex] and [latex]z[/latex] to represent numbers and [latex]A,\text{ }B,[/latex] and [latex]C[/latex] to represent sets.

[latex]\begin{array}{lllll}h\text{ is }f\text{ of }a\hfill & \hfill & \hfill & \hfill & \text{We name the function }f;\text{ height is a function of age}.\hfill \\ h=f\left(a\right)\hfill & \hfill & \hfill & \hfill & \text{We use parentheses to indicate the function input}\text{. }\hfill \\ f\left(a\right)\hfill & \hfill & \hfill & \hfill & \text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\hfill \end{array}[/latex]

Remember, we can use any letter to name the function; the notation [latex]f\left(a\right)[/latex] shows us that height, [latex]h,[/latex] depends on age, [latex]a.[/latex] The value [latex]a[/latex] must be put into the function [latex]f[/latex] to get the height. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example [latex]f\left(a+b\right)[/latex] means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

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. Assume that the domain does not include leap years.

Show Solution

The number of days in a month is a function of the name of the month, so if we name the function [latex]f,[/latex] we write [latex]\text{days}=f\left(\text{month}\right)[/latex] or [latex]d=f\left(m\right).[/latex] The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

Figure 5

For example, [latex]f\left(\text{March}\right)=31,[/latex] because March has 31 days. The notation [latex]d=f\left(m\right)[/latex] reminds us that the number of days, [latex]d[/latex] (the output), is dependent on the name of the month, [latex]m[/latex] (the input).

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. However, most of the functions we will work with in this book will have numbers or algebraic expressions as inputs and outputs.

Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

Table 3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function [latex]f[/latex] where [latex]D=f\left(m\right)[/latex] identifies months by an integer rather than by name.

Table 4 defines a function [latex]Q=g\left(n\right).[/latex] Remember, this notation tells us that [latex]g[/latex] is the name of the function that takes the input [latex]n[/latex] and gives the output [latex]Q\text{ .}[/latex]

Table 5 displays the age of children in years and their corresponding heights. 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. and 42 in.

When a table represents a function, corresponding input and output values can also be specified using function notation.

The function represented by Table 6 can be represented by writing

[latex]f\left(2\right)=1,\text{ }f\left(5\right)=3,[/latex] and [latex]f\left(8\right)=6.[/latex]

Similarly, the statements

[latex]g\left(-3\right)=5,\text{ }g\left(0\right)=1,[/latex] and [latex]g\left(4\right)=5[/latex]

represent the function in Table 7.

Table 8 cannot be expressed in a similar way because it does not represent a 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. For example, the function [latex]f\left(x\right)=5-3{x}^{2}[/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

In Example 6, you worked with the expression below:

[latex]\begin{align*}\frac{f(a+h)-f(a)}{h}\\\end{align*}[/latex]

This is called the difference quotient. In Calculus, we use the difference quotient to develop an important concept called the derivative. You should work to become very comfortable with the difference quotient. We will work with it more in later sections of this chapter.

Example 8:  Solving Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p,[/latex] solve for [latex]h\left(p\right)=3.[/latex]

Show Solution

[latex]\begin{align*}h\left(p\right)&=3&&\text{ }\\{p}^{2}+2p&=3&&\text{Substitute the original function.}\\{p}^{2}+2p-3&=0&&\text{Subtract 3 from each side}.\\\left(p+3\text{)(}p-1\right)&=0&&\text{Factor}.\end{align*}[/latex]

 

If [latex]\left(p+3\right)\left(p-1\right)=0,[/latex] either [latex]\left(p+3\right)=0[/latex] or [latex]\left(p-1\right)=0[/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[/latex] in each case.

[latex]\begin{array}{ll}\left(p+3\right)=0,\hfill & p=-3\hfill \\ \left(p-1\right)=0,\hfill & p=1\hfill \end{array}[/latex][latex]\\[/latex]

This gives us two solutions. The output [latex]h\left(p\right)=3[/latex] when the input is either [latex]p=1[/latex] or [latex]p=-3.[/latex] We can also verify by graphing as in Figure 6. The graph verifies that [latex]h\left(1\right)=h\left(-3\right)=3[/latex] and [latex]h\left(4\right)=24.[/latex]

Figure 6

Evaluating Functions Expressed in Formulas

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. For example, the equation [latex]2n+6p=12[/latex] expresses a functional relationship between [latex]n[/latex] and [latex]p.[/latex] We can rewrite it to decide if [latex]p[/latex] is a function of [latex]n.[/latex]

Finding Function Values from a Graph

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).

Example 12:  Reading Function Values from a Graph

Given the graph in Figure 7,

1. Evaluate [latex]f\left(2\right).[/latex]
2. Solve [latex]f\left(x\right)=4.[/latex]

Figure 7

Show Solution

1. To evaluate [latex]f\left(2\right),[/latex] locate the point on the curve where [latex]x=2,[/latex] then read the y-coordinate of that point. The point has coordinates [latex]\left(2,1\right),[/latex] so [latex]f\left(2\right)=1.[/latex] See Figure 8
   

   Figure 8
2. To solve [latex]f\left(x\right)=4,[/latex] we find the output value [latex]4[/latex] on the vertical axis. Moving horizontally along the line [latex]y=4,[/latex] we locate two points of the curve with output value [latex]4:[/latex] [latex]\left(-1,4\right)[/latex] and [latex]\left(3,4\right).[/latex] These points represent the two solutions to [latex]f\left(x\right)=4:[/latex] [latex]-1[/latex] or [latex]3.[/latex] This means [latex]f\left(-1\right)=4[/latex] and [latex]f\left(3\right)=4,[/latex] or when the input is [latex]-1[/latex] or [latex]\text{3,}[/latex] the output is [latex]\text{4}\text{.}[/latex] See Figure 9.
   

   Figure 9

Try it #8

Using Figure 7, solve [latex]f\left(x\right)=1.[/latex]

Show Solution

[latex]x=0[/latex] or [latex]x=2[/latex]

Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in Figure 10, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.

Figure 10

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. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in Table 12.

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.

To visualize this concept, let’s look again at the two simple functions sketched in Figure 1(a) and Figure 1(b). The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[/latex] and [latex]r[/latex] both give output [latex]n.[/latex] The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

Using the Vertical Line Test

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.

Very often graphs name the input value [latex]x[/latex] and the output value [latex]y,[/latex] and we say [latex]y[/latex] is a function of [latex]x,[/latex] or [latex]y=f\left(x\right)[/latex] when the function is named [latex]f.[/latex] The graph of the function is the set of all points [latex]\left(x,y\right)[/latex] in the plane that satisfies the equation [latex]y=f\left(x\right).[/latex] 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. For example, the black dots on the graph in Figure 11 tell us that [latex]f\left(0\right)=2[/latex] and [latex]f\left(6\right)=1.[/latex] However, the set of all points [latex]\left(x,y\right)[/latex] satisfying [latex]y=f\left(x\right)[/latex] is a curve. The curve shown includes [latex]\left(0,2\right)[/latex] and [latex]\left(6,1\right)[/latex] because the curve passes through those points.

Figure 11

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 can have only one output value for each input value. See Figure 12.

Figure 12

The second and third graphs in Figure 12 are not functions because the input value represented by the dotted line has two output values in each case.  The two output values are the [latex]y[/latex] values where each dotted line intersects the solid line.  This contradicts the definition of a function. Remember, a function is a rule where each input is mapped to exactly one output.

Example 14:  Applying the Vertical Line Test

Which of the graphs in Figure 13 represent(s) a function [latex]y=f\left(x\right)?[/latex]

Figure 13

Show Solution

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 13. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at x-values between -3 and 3, a vertical line would intersect the graph at more than one point, as shown in Figure 14.  This indicates that each of these inputs gets mapped to two different outputs.  This contradicts the definition of a function, since we know a function maps each input to exactly one output.

Figure 14

Try it #11

Does the graph in Figure 15 represent a function?

Figure 15

Show Solution

Yes.  Any vertical line will only pass through the graph once.  This confirms the definition of one-to-one, since each output corresponds to only one input.

Using the Horizontal Line Test

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.  Each intersection along the horizontal line represents an x-value with the same output which contradicts the definition of one-to-one which states that each  output value must be unique for the function to be one-to-one.

Example 15:  Applying the Horizontal Line Test

Consider the functions shown in Figure 13 (a) and Figure 13(b). Are either of the functions one-to-one?

Show Solution

The function in Figure 13(a) is not one-to-one. The horizontal line shown in Figure 16 intersects the graph of the function at two points.  These two points have the same output value but different input values.  Remember that the definition of a one-to-one function is that each output value corresponds to exactly one input value.  Therefore, when a horizontal line intersects a graph at more than one point, we have a contradiction to this definition, and the function cannot be a one-to-one function.

Figure 16

The function in Figure 13(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.  This means that each output is associated with only one input value, satisfying the definition of a one-to-one function.

Identifying Basic Toolkit Functions

In this text, we will be exploring functions including 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. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable.

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. The graphs and sample table values are included with each function shown in Table 13.

Table 13

Toolkit Functions
Name	Function	Graph
Constant	[latex]f\left(x\right)=c,[/latex] where [latex]c[/latex] is a constant
Identity	[latex]f\left(x\right)=x[/latex]
Absolute value	[latex]f\left(x\right)=|x|[/latex]
Quadratic	[latex]f\left(x\right)={x}^{2}[/latex]
Cubic	[latex]f\left(x\right)={x}^{3}[/latex]
Reciprocal	[latex]f\left(x\right)=\frac{1}{x}[/latex]
Reciprocal squared	[latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex]
Square root	[latex]f\left(x\right)=\sqrt{x}[/latex]
Cube root	[latex]f\left(x\right)=\sqrt[3]{x}[/latex]