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Here are a few recommended readings before getting started with this lesson.
A function is a relation in which each input is assigned to exactly one output. The set of all possible inputs is called the domain of the function and the set of all possible outputs is called the range. If $x$ represents the inputs and $y$ the outputs of a function, it is often said that $y$ is a function of $x$
or that $y$ depends on $x.$
$y=f(x)$
Determining Whether a Relation Is a Function  

If represented as  Use 
a set of coordinates or a table of values  a mapping diagram 
a graph in the coordinate plane  the vertical line test 
If the relation is given as a vertical table, the inputs are the values in the first column.
If the relation is given as a vertical table, the outputs are the values in the second column.
Draw an arrow from each input to its corresponding output.
Check if there are multiple arrows with the same starting input. In other words, identify inputs that have multiple outputs.
Here, the input $4$ has two different outputs, $3$ and $5.$ By definition, for a relation to be a function, every input must be assigned to exactly one output. As such, the given relation is not a function.
Kevin's teacher Maya is organizing a class trip to the Aquatic Wonders World Aquarium. She examines the aquarium's website and observes the number of visitors recorded over the past week.
Day, $x$  $1$  $2$  $3$  $4$  $5$  $6$  $7$ 

Number of Visitors, $N$  $1300$  $1500$  $1800$  $1300$  $1900$  $2500$  $1900$ 
The numbers in the first row represent the days of the week, with the number $1$ indicating Monday.
Now each element in the first column can be connected to its corresponding element in the second column with an arrow.
Notice that the same number of visitors appears on certain days, but there are no two visitor counts on a single day. In other words, each input is assigned to exactly one output. Therefore, this relation is a function.
The vertical line test is a graphical method to determine whether a given relation is a function. For example, consider the following relations.
Relation I  Relation II  Relation III 

$(y−2)(y+1)=x$  $y=x(x+1)(x−2)$  $xy 2.252 1.750 12 11 01 10.75 20.5 $

Draw the relation on the coordinate plane. The first two relations can be drawn using a graphing calculator or other mathematical software.
Draw vertical lines at different places through the coordinate plane. If one of the lines intersects the graph more than once, the relation is not a function. Conversely, if no vertical line cuts the graph more than once, the relation is a function.
Notice that $ℓ_{3}$ cuts the first graph at two different points. The line $m_{2}$ also passes through two different points. This means that neither Relation I nor Relation III is a function. However, all of the vertical lines drawn over Relation II only intersect the graph one time at most. Because of this, Relation II is a function.
Keep in mind that before stating whether a relation is a function, the vertical lines drawn have to cover the entire domain to ensure that no vertical line cuts the graph more than once.
If a relation is not a function, it is because there are multiple $y$values corresponding to the same $x$value. Therefore, the graph of such relation would show at least two points directly above the other.
Points with the same $x$value belong to the same vertical line.
Maya and her students are about to start their visit. Their journey begins with a short seminar that gives cool facts about the aquarium. During the presentation, a graph that shows the fish population over the past $8$ months was presented.
However, on the screen in front of Kevin, the axes were positioned in reverse. Consider the shape of the graph. It appears quite different.
Which of the two graphs represents a function?Use the vertical line test to determine whether the graphs represent functions.
The following applet displays a relation as a set of ordered pairs, a group of coordinate points in the coordinate plane, or a curve in the coordinate plane. Determine whether the relation is a function or not.
$y$ equals $f$ of $x.$Equations that are functions can be written using function notation.
Besides $f,$ other letters such as $g$ or $h$ can be used to name the function. Similarly, letters other than $x$ can name the independent variable.
Function  Analysis  Domain 

$f(x)=3x$  Multiplication by $3$ is defined for all real numbers.  All real numbers 
$g(x)=x $  Square roots are not defined for negative numbers.  All nonnegative numbers — that is, $x≥0$ 
$h(x)=x1 $  Division by zero is undefined.  All real numbers except $0$ — that is, $x =0$ 
The range of a function is the set of all $y$values, or outputs, of the function. The range of a function depends on both the domain and the function itself. For example, consider the following functions and their defined domains.
Function  Domain 

$f(x)=2x$  All integers 
$g(x)=x_{2}$  All real numbers 
$h(x)=4$  All real numbers 
By analyzing the definition of each function along with the given domains, the ranges can be determined.
Function  Domain  Analysis  Range 

$f(x)=2x$  All integers  The function takes any integer input and produces an output that is an even number, as each input is multiplied by $2.$  All even numbers 
$g(x)=x_{2}$  All real numbers  The function takes any real number input and produces an output that is a nonnegative number, as each input is squared.  All nonnegative numbers — that is, $y≥0$ 
$h(x)=4$  All real numbers  The function takes any real number input and sends it to $4.$  Only the number $4$ — that is, the range is ${4}$ 
Kevin, feeling thirsty after the seminar, buys water from a nearby vending machine. There are six types of products sold in the vending machine: orange juice, water, chocolate, donuts, sandwiches, and pizza.
Notice that the buttons on the vending machine are labeled with the first letters of the food items. When a button is pressed, the machine will give the corresponding food.
Is it a function? Yes
Range: ${$Chocolate, Donut, Orange Juice, Pizza, Sandwich, Water$}$
Now each element in the first column can be connected with an arrow to its corresponding element in the second column.
Notice that each element of the input set is paired with exactly one element of the output set, so this is onetoone mapping. Therefore, this relation is a function. This conclusion makes sense because it is not logical that pressing a button would result in two different snacks being dispensed.
Maya and her students start their tour of the aquarium with a visit to the famous dolphins. The students excitedly gather around the dolphin exhibit and watch the playful animals swimming in the water.
The guide shares interesting facts about these intelligent creatures, capturing the students' attention and sparking their curiosity about the animals in the aquarium. Kevin notes this information as a set of ordered pairs.One way to understand functions is to think of functions as machines. Inputs are like raw materials that go through the processing stage of the function. Outputs are the final product. In this applet, four preset inputs are available. The machine specifically processes numbers between $100$ and $100$ as materials. Try plugging in a few values and see the outcomes!
The dolphin trainer claims that the dolphin in front of him eats $15$ kilograms of fish per day.
Graph:
$d$  $15d$  $k$ 

$1$  $k=15(1)$  $15$ 
$3$  $k=15(3)$  $45$ 
$5$  $k=15(5)$  $75$ 
$7$  $k=15(7)$  $105$ 
Now plot the ordered pairs $(d,k)$ as points in a coordinate plane.
Finally, connect the points with a line. Note that negative numbers have no meaning in the context of the problem, so the graph will only be in the first quadrant.