# Function Notation

### Learning Outcomes

- Write functions using algebraic notation
- Use the vertical line test to determine whether a graph represents a function

*f”*for function. If you put

*x*into the box,

*f*(

*x*)

*,*comes out. Mathematically speaking,

*x*is the input, or the “independent variable,” and

*f*(

*x*) is the output, or the “dependent variable,” since it depends on the value of

*x*. [latex]f(x)=4x+1[/latex] is written in function notation and is read “

*f*of

*x*equals [latex]4x[/latex] plus [latex]1[/latex].” It represents the following situation: A function named

*f*acts upon an input,

*x,*and produces

*f*(

*x*) which is equal to [latex]4x+1[/latex]. This is the same as the equation [latex]y=4x+1[/latex]. Function notation gives you more flexibility because you do not have to use

*y*for every equation. Instead, you could use

*f*(

*x*) or

*g*(

*x*) or

*c*(

*x*). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.

## Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationship so that we can understand it, use it, and possibly even program it into a computer. There are various ways of representing functions. A standard**function notation**is one representation that facilitates working with functions. Now you try it.

### Example

Represent height as a function of age using function notation.Answer: To represent "height as a function of age," we start by identifying the variables: [latex]h[/latex] for height and [latex]a[/latex] for age. [latex-display]\begin{array}{ccc}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 ''<em>f</em> of a."}\hfill \end{array}[/latex-display] Note: We can use any letter to name the function; the notation [latex]h=f\left(a\right)[/latex] shows us that [latex]h[/latex] depends on, or is a function of, [latex]a[/latex]. The value [latex]a[/latex] must be put into the function [latex]f[/latex] to get a result (height). The parentheses indicate that age is the input for the function; they do not indicate multiplication.

### Example

- Write the formula for perimeter of a square, [latex]P=4s[/latex], as a function.
- Write the formula for area of a square, [latex]A=l^{2}[/latex], as a function.

Answer:

- Name the function [latex]P[/latex]. [latex]P[/latex] is a function of the length of the sides, [latex]s[/latex]. Perimeter, as a function of side length, is equal to [latex]4[/latex] times side length. [latex]P(s)=4s[/latex]
- Name the function [latex]A[/latex]. Area, as a function of the length of the sides, is equal to the length squared. [latex]A(l)=l^{2}[/latex].

*a*and

*b*, and the result is the input for the function [latex]f[/latex]." The operations must be performed in this order to obtain the correct result.

### Function Notation

The notation [latex]y=f\left(x\right)[/latex] defines a function named [latex]f[/latex]. This is read as "[latex]y[/latex] is a function of [latex]x[/latex]." The letter [latex]x[/latex] represents the input value, or independent variable. The letter*[latex]y[/latex]*or [latex]f\left(x\right)[/latex], represents the output value, or dependent variable.

### Example

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.Answer: 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. For example, [latex]f\left(\text{March}\right)=31[/latex], because March has [latex]31[/latex] 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).

### Example

A function [latex]N=f\left(y\right)[/latex] gives the number of police officers, [latex]N[/latex], in a city in year [latex]y[/latex]. What does [latex]f\left(2005\right)=300[/latex] represent?Answer: When we read [latex]f\left(2005\right)=300[/latex], we see that the input year is 2005. The value for the output, the number of police officers, [latex]N[/latex], is 300. Remember, [latex]N=f\left(y\right)[/latex]. The statement [latex]f\left(2005\right)=300[/latex] tells us that in the year 2005 there were [latex]300[/latex] police officers in the city.

## Graphs of Functions

When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the*x*-axis and the dependent value is plotted on the

*y*-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as [latex]y = f(x)[/latex], or y is a function of [latex]x[/latex], we would write ordered pairs [latex](x, f(x))[/latex] using function notation instead of [latex](x,y)[/latex] as you may have seen previously. We can identify whether the graph of a relation represents a function because for each

*x*-coordinate there will be exactly one

*y*-coordinate. When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of

*x*. If, on the other hand, a graph shows two or more intersections with a vertical line, then an input (

*x*-coordinate) can have more than one output (

*y*-coordinate), and

*y*is not a function of

*x*. Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the “vertical line test.” You try it.

### Example

Use the vertical line test to determine whether the relation plotted on this graph is a function.Answer:
This relationship cannot be a function, because some of the *x*-coordinates have two corresponding *y*-coordinates. We can see this in the graph below because there are vertical lines that pass through the graph in two different places.

### Example

Consider the ordered pairs [latex]\{(−1,3),(−2,5),(−3,3),(−5,−3)\}[/latex] plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.

Answer:

Drawing vertical lines through each point results in each line only touching one point. This means that none of the *x*-coordinates have two corresponding *y*-coordinates, so this is a function.

*x*-coordinate has multiple

*y*-coordinates. This relation is not a function. In the following video, we show another example of determining whether a graph represents a function using the vertical line test. https://youtu.be/5Z8DaZPJLKY

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- Ex: Function Notation Application Problem.
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution. - Function Notation Application.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - College Algebra.
**Provided by:**OpenStax**Located at:**https://cnx.org/contents/[email protected]:1/Preface.**License:**CC BY: Attribution.**License terms:**Download for free at : http://cnx.org/contents/[email protected]:1/Preface. - Ex: Determine if a Table of Values Represents a Function.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.