Sign In
| 20 Theory slides |
| 12 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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 |
Start by writing all the inputs of the relation in a column. The inputs are the first components of each pair if the relation is given as a set of coordinates. {( 4,3),( 0,-5),( -3,3),( 8,0),( 4,-5)} Notice that the number 4 is an input for two different points. In the mapping diagram, it is written only once because multiple arrows can be drawn from it. Therefore, the inputs for the given relation are 4, 0, -3, and 8.
If the relation is given as a vertical table, the inputs are the values in the first column.
Write all the outputs in a second column. When the relation is given as a set of coordinates, the outputs are the second components of each pair. {(4, 3),(0, -5),(-3, 3),(8, 0),(4, -5)} The number - 5 is an output for two different pairs. In the mapping diagram, it is written only once because multiple arrows can be drawn to it. For the given relation, the outputs are 3, -5, and 0.
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) | |c|c|c|c|c|c|c|c| x & -2.25 & -1.75 & -1 & -1 & 0 & 1 & 2 y & -2 & 0 & 2 & -1 & 1 & -0.75 & 0.5 |
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.
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.
Function notation is a special way to write functions that explicitly shows that y is a function of x — in other words, that y depends on x. Function notation is symbolically expressed as y=f(x) and read y equals f of x.
Equations that are functions can be written using function notation.
ccc
Equation & & Function Notation [1ex]
y=-5x+4 & & f(x) = -5x+4
Notice that y has been replaced by f(x). In function notation, x represents an element of the domain and f(x) represents the element of the range that corresponds to x. When written in function notation, the expression that describes how to convert an input into an output — the right-hand side expression — is called the function rule.
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.
To interpret an equation given in function notation, it is necessary to understand what both sides of f(x)=k mean. For example, consider the following equation. f( 3) = 12 Here, f( 3) denotes that the function's input is x= 3 and that 12 is the output corresponding to this input. f( 3) = 12 ⇓ The output offwhenx= 3 is 12. Now, consider a different scenario. Let w(t)=200t be a function that describes the number of words Kevin reads in t minutes. The following statements are true for this function. w(4) and w(t)=900 Here, w(4) is the number of words that Kevin reads in 4 minutes and can be found by evaluating the function for t=4. However, the input is not a particular number in the second statement. In such cases, the statement can be interpreted as a question. w(t)=900 ⇓ For which value oft is the output equal to900? Based on the context, the second statement asks how many minutes it takes Kevin to read 900 words. To find this value of t, the equation w(t)=900 has to be solved for t.
The domain of a function is the set of all x-values, called inputs, for which the function is defined. As an example, consider the following functions. f(x) & = 3x [0.2cm] g(x) & = sqrt(x) [0.3em] h(x) & = 1/x Their domains can be written by analyzing the definition of each function.
Function | Analysis | Domain |
---|---|---|
f(x) = 3x | Multiplying by 3 is defined for all real numbers. | All real numbers |
g(x) = sqrt(x) | Square roots are not defined for negative numbers. | All non-negative numbers — that is, x≥ 0 |
h(x) = 1/x | Dividing 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, called outputs, of the function. The range 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 |
The ranges of each function can be determined by analyzing the definition of each function along with the given domains.
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 non-negative number, as each input is squared. | All non-negative 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 one-to-one 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.
When sorted in alphabetical order, the domain and range of the function can be written as follows. Domain: &{C,D,O,P,S,W } [0.8em] Range: & { l Chocolate,Donut,Orange Juice, Pizza,Sandwich,Water }
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. K = {(1.9,120),(2.3,138),(2.5,140),(2,138) } Here, every pair of numbers represents the length of a dolphin in meters and its weight in kilograms. For example, the pair (2,138) indicates that a dolphin is 2 meters long and weighs 138 kilograms.
{(1.9,120),(2.3,138),(2.5,140),(2,138) } It is stated that the first coordinates indicate the lengths of dolphins, while the second coordinates represent their weights. This means that the x-axis corresponds to length and the y-axis corresponds to weight. The points can now be plotted.
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!
In the context of functions, the input is often referred to as the independent variable because it can be chosen arbitrarily from the domain. Conversely, the output is called the dependent variable because its value depends on the value of the independent variable. For instance, if the price of oranges is $ 2.50 per pound, the total cost is determined by the product of the unit price and the weight in pounds. ccccc Cost & & Unit Price & & Weight [0.4em] y & = & 2.50 & * & x
As shown, the total cost of oranges depends on how many pounds of fruit are purchased. Therefore, the cost of oranges y is the dependent variable and the number of pounds purchased x is the independent variable.The dolphin trainer claims that the dolphin in front of him eats 15 kilograms of fish per day.
Graph:
Independent Variable: & d Dependent Variable: & k Therefore, the total amount of fish eaten by the dolphin k can be written as a function of d. It will be equal to the product of 15 and the number of days d. Equation k = 15 d This equation can also be expressed using function notation. In this case, it would be appropriate to use the notation k(d) as it shows that k is a function of d. Function Notation k(d) = 15d Next, make a table of values to help graph the function. For example, evaluate the function when d is 1, 3, 5, and 7.
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.
Dylan creates a table of values for a function with the intention of graphing it. The diagram shows his work.
Let's take a look at the table Dylan created. Notice that each output is 2 greater than its respective input.
Input, x | Output, y |
---|---|
- 5 | - 3 = - 5 + 2 |
- 2 | 0 = - 2+ 2 |
0 | 2 = 0+ 2 |
1 | 3 = 1+ 2 |
We can conclude that the table represents the function f(x) = x+2. The ordered pairs ( - 5, - 3), ( - 2, 0), ( 0, 2), and ( 1, 3) lie on this line. This means that Statement II is true.
We can also conclude that Dylan did not draw the function correctly because the graph he drew is different than the graph of f(x) = x+2. This means that Statement I is false. Let's take a look at the graph he drew.
It looks like Dylan drew his graph by using the input values as the y-coordinates and the output values as the x-coordinates for the points. As a result, the graph he drew is the graph of f(x) = x - 2.
Input | Output |
---|---|
- 3 | - 5 = - 3 -2 |
0 | - 2 = 0-2 |
2 | 0 = 2 -2 |
3 | 1 = 3-2 |
We found that Statements II and III are true.
In a chemistry experiment, LaShay tracks how the temperature of a substance changes over time. She notes down the numbers as ordered pairs (time, temperature, with the time in minutes and the temperature in degrees Celsius. { (0,20),(10,30), (15, 50),(20, 100), (30, 100) } The experiment is terminated at the 30th minute. At the end of the experiment, LaShay draws the following graph.
Let's take a look at the graph LaShay drew.
As time passes, the temperature of the substance changes. This means that the temperature depends on time. From this, we know that the temperature of the substance T is the dependent variable and the time in minutes t is the independent variable. Each t-value seems to have exactly one T-value assigned to it, which we can confirm using the vertical line test. We know from this that Statement I is true. lThe temperature of the substance is a function of time. ✓ Now let's determine what values of the domain and range make sense for the function. The domain is the set of possible values of t.
The experiment lasts 30 minutes, so the numbers between 0 and 30 will form the domain. Since time is a continuous quantity, all real numbers between 0 and 30 are included in the domain. We can write it as a compound inequality. Domain: 0 ≤ t ≤ 30 This means that Statement II is also true. Now let's find the range. The range of the function is the set of possible values of T.
Since temperature can also be thought of as a continuous quantity, all real numbers between 20 and 100 are included in the range. Range: 20≤ T ≤ 100 As a result, only the first two statements are correct.
Jordan approximates the area of a circle using the function A(r) = 3 r^2, where r is the radius of the circle in meters.
Which of the following is the graph of the function?
We know that the area of a circle changes as its radius changes. This holds true for the function Jordan uses to approximate the area of the circle. A(r) = 3 r^2 Here, the radius r of the circle in meters is the independent variable and its area A(r) in square meters is the dependent variable. Since r is a length, it cannot be negative. Because of this, we will use only non-negative values for r and find some ordered pairs we can use to graph the function.
r | 3r^2 | A(r) = 3r^2 |
---|---|---|
0 | 3( 0)^2 | 0 |
1 | 3( 1)^2 | 3 |
2 | 3( 2)^2 | 12 |
3 | 3( 3)^2 | 27 |
4 | 3( 4)^2 | 48 |
For non-negative r-values, we get only positive A-values, so let's restrict our graph to the first quadrant. In our coordinate plane, the horizontal axis represents r-values, while the vertical axis represents the A-values. Let's plot the ordered pairs ( 0, 0), ( 1, 3), ( 2, 12), ( 3, 27), and ( 4, 48). Then we can connect the points as the points between them with a smooth curve.
This corresponds to graph D.
Recall that r represents the inputs of the function. This means that the domain of the function is the set of possible values of r.
We can see that r can take any real number greater than or equal to 0. Let's write this as an inequality. Domain: r ≥ 0
The outputs of the function are represented by A. The range is the set of possible values of A.
The function rule results in non-negative real number outputs for non-negative real number inputs. Therefore, we can write the range as follows. Range: A ≥ 0
Every month, Diego tracks the number of smartphones his company has in stock. The table shows the weekly stock.
Weeks Passed, w | 0 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|
Smartphone Inventory, s | 240 | 200 | 160 | 120 | 80 |
Which is the graph of ordered pairs in the table?
Let's take a look at the table that shows the relationship between the number of smartphones in stock and weeks.
Weeks Passed, w | 0 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|
Smartphone Inventory, s | 240 | 200 | 160 | 120 | 80 |
As the weeks pass, the number of phones in stock decreases. In this case, the dependent variable is the number of phones in stock s, while the independent variable is the number of weeks w. Independent Variable:& w Dependent Variable: & s From this, we know that the table represents the following ordered pairs written in the form ( w, s). ( 0, 240), ( 2, 200), ( 4, 160), ( 6, 120),( 8, 80) Let's plot these points on a coordinate plane. We will place the inputs w along the horizontal axis and the outputs s along the vertical axis.
This graph corresponds to option C.
We want to determine the timing for ordering more smartphones. Diego will order more smartphones when there are no phones left in stock. We can find this timing by following the pattern in the graph we drew in Part A.
We can see that moving 2 units to the right along the graph is followed by moving down 40 units. Let's repeat this pattern a couple of times until we reach the horizontal axis where the number of smartphones is 0.
This means that after 12 weeks, there will be no smartphones remaining in stock! Diego should order more phones after 12 weeks.
Alternatively, we could draw a line that passes through the points in the graph since all points follow the same pattern.
The point where the line and the horizontal axis intersect indicates the answer.
A swimming pool can hold 120 000 liters of water. Water is being pumped into the pool at a rate of 8000 liters per hour. The total volume of water V is a function of time in hours t.
We know that water is being pumped into a swimming pool at a rate of 8000 liters per hour. The amount of water in the pool changes over time. Since the total amount of water inside the pool V is a function of the time in hours t, the dependent variable is V and the independent variable is t. Dependent Variable: & V Independent Variable: & t
Let's write the function rule for V. We know that water is being pumped into the pool at a rate of 8000 liters per hour. Therefore, after t hours, the amount of water in the pool will be 8000 times t.
V = 8000* t ⇔ V(t) = 8000* t
This function represents the total amount of water in the pool!
We are asked to determine what values make sense for the domain and range of the function V(t). Let's find the domain first!
The domain is the set of possible values of t. The values of t represent the amount of time that water is pumped into the pool. Since t is time, negative values of t do not make sense. This means that our domain should be greater than or equal to 0. t ≥ 0 However, we cannot pump water forever because at some point the pool will reach its maximum capacity. Let's try to find the time t at which the total amount of water inside the pool V(t) reaches its maximum, 120 000 liters. In other words, we need to solve V(t) = 120 000 for t. Let's use the function we wrote in Part B, V(t)=8000t.
We found that t=15, which means the pool will fill in 15 hours. Therefore, values of t greater than 15 do not make sense in this context. The domain of the function will be at most 15. t≤ 15 By combining the two inequalities, we can express the domain as compound inequality. Domain t ≥ 0 and t ≤ 15 ⇕ 0≤ t≤ 15
The range is the set of possible values of V, the total amount of water in the pool. We cannot have a negative amount of water in the pool. Initially, at t=0, the pool contains no water. We are given that the pool can hold up to 120 000 liters of water. Therefore, V is greater than or equal to 0 and less than or equal to 120 000. Range 0≤ V ≤ 120 000 As we can see, the correct option is A.
To find the total amount of water in the pool after 4 hours, we need to evaluate the function at t=4. Let's substitute t=4 into the function and evaluate the right-hand side.
After 4 hours, the pool will have 32 000 liters of water.