[latex]f(x)=4x+1[/latex] is written in function notation and reads as follows: “[latex]f[/latex] of [latex]x[/latex] equal to [latex]4x[/latex] plus 1”. It represents the following situation: A function named [latex]f[/latex] acts on an input, [latex]x[/latex], and creates [latex]f(x)[/latex], which is equal to [latex]4x+1[/latex]. This is the same as the equation of [latex]y=4x+1[/latex]. We have only listed 4 of them once because there is no need to list them every time they appear in the section. [latex]begin{array}{ccc}htext{ is }ftext{ of }ahfill & hfill & hfill & text{We call the function }f;text{ height is a function of age}.hfill h=fleft(aright)hfill & hfill & hfill & hfill & hfill & text{We use parentheses to display the input function}text{. } hfill fleft aright)hfill & hfill & hfill & hfill & hfill & hfill & text{We call the function }f;text{ the expression is read as “}ftext{ by }atext{“}. hfill end{array}[/latex] You may also be prompted to evaluate a function for multiple values, as shown in the following example. Note that in the first table below, where the entry is “Name” and the output is “Age”, each entry corresponds exactly to an output. This is an example of a function. Functions can also be evaluated for negative values of [latex]x[/latex]. Observe the rules for integer operations. By organizing the ordered pairs in a table, you can determine whether this relationship is a function. By definition, the inputs of a function have only one output.

You can simply apply what you already know about evaluating expressions to evaluate a function. It is important to note that the parentheses that are part of the function notation do not multiply. The notation [latex]f(x)[/latex] does not mean [latex]f[/latex] multiplied by [latex]x[/latex]. Instead, the notation means “[latex]f[/latex] of [latex]x[/latex]” or “the function of [latex]x[/latex]” To evaluate the function, take the value specified for [latex]x[/latex] and replace that value with [latex]x[/latex] in the expression. Let`s look at some examples. To represent “height is a function of age”, we first identify the descriptive variables [latex]h[/latex] for height and [latex]a[/latex] for age. The range is the list of outputs for the relationship, they entered in second place in the ordered pair. You read this problem like this: “Since [latex] f[/latex] of [latex]x[/latex] is equal to [latex]4x[/latex] plus one, find [latex]f[/latex] of 2.” Although the notation and formulation are different, the process of evaluating a function is the same as that of an equation: in both cases, you replace x with 2, multiply it with 4, and add 1 to get 9.

In a function and an equation, an input of 2 gives an output of 9. You can also call the machine [latex]f[/latex] for the function. If you enter [latex]x[/latex] in the box, [latex]f(x)[/latex] appears. Mathematically, [latex]x[/latex] is the input or “independent variable” and [latex]f(x)[/latex] is the output or “dependent variable” because it depends on the value of [latex]x[/latex]. Let`s look at our examples to determine whether relationships are functions or not, and under what circumstances. Remember that a relationship is a function if there is only one output for each input. The domain describes all the entries, and we can use set notation with parentheses{} to create the list. Algebra gives us a way to explore and describe relationships. Imagine throwing a ball directly into the air and watching it rise to its climax before it falls back into your hands. Over time, the height of the ball changes. There is a relationship between the time that has elapsed since the throw and the height of the ball. In mathematics, a correspondence between variables that change together (e.B.

time and altitude) is called a relationship. Some, but not all, relationships can also be described as functions. We list all input values as domain. Input values are classically displayed first in the ordered pair. Once we have established that a relationship is a function, we need to see and define functional relationships so that we can understand and use them, and sometimes to be able to program them in computers. There are several ways to represent functions. A standard function notation is a representation that makes it easier to use functions. A function [latex]N=fleft(yright)[/latex] indicates the number of police officers, [latex]N[/latex], in a city of the year [latex]y[/latex].

What does [latex]fleft(2005right)=300[/latex] mean? 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. To determine whether this is a function, you can reorder the information by creating a table. We can use any letter to name the function; the notation [latex]hleft(aright)[/latex] shows us that [latex]h[/latex] depends on [latex]a[/latex]. The value [latex]a[/latex] must be entered in the function [latex]h[/latex] to obtain a result. Parentheses indicate that age has entered the function; they do not indicate multiplication. This would facilitate the graphical representation of the two functions on the same graph, without confusion about the variables. Represent size as a function of age using function notation. Note that the entries for a function do not have to be numbers. Feature entries can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the features we will be working with in this book have numbers like inputs and outputs. In the following example, you evaluate a function for an expression.

So here you`re going to replace and simplify the whole expression in for [latex]x[/latex]. The notation [latex]y=fleft(xright)[/latex] defines a function called [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 y or [latex]fleft(xright)[/latex] represents the output value or dependent variable. Function notation gives you more flexibility because you don`t need to use [latex]y[/latex] for every equation. Instead, you can use [latex]f(x)[/latex] or [latex]g(x)[/latex] or [latex]c(x)[/latex]. This can be a useful way to distinguish equations from functions when you`re dealing with more than one at a time. The first value in a relationship is an input value, and the second value is the output value.

A function is a specific type of relationship where each input value has one and only one output value. An input is the independent value and the output value is the dependent value because it depends on the value of the input. There is a name for the set of input values and another name for the set of output values for a function. . 