The concept of a function

Look at the following mapping diagram (Venn diagram) relating a group of friends to their favorite t-shirt color. Does this relation represent a function? Explain why.

One of the golden rules for deciding whether a graph represents a function is to check that for each value of the independent variable $x$ there is exactly one value of the dependent variable $y$. State which of the following graphs is NOT a function and briefly justify your answer.

Indicate which of the following relations are not functions and explain the reason based on the definition of a function:

1. To each person in your class, we assign their date of birth.

2. To each natural number, we assign its factors.

3. To each real number, we assign its positive square root.

4. To each mother, we assign the names of her children.

5. To each product in a store, we assign its price in pesos.

To hire an electric scooter, a company charges $90$ pesos as an unlock fee and $12$ pesos per minute of use. We want to analyse the relation between the time of use and the total hire cost.

Identify the variables of this relation:

• Independent variable ($x$): ________

• Dependent variable ($y$): ________

Complete the following table of values knowing that the function associates each natural number with three times its value minus one. That is, the algebraic expression is $y = 3x - 1$.

Given the function $f(x) = x^2 - 3$, calculate the value of the dependent variable for the following values of $x$:

1. a) If $x = 0 \rightarrow y = $
2. b) If $x = 2 \rightarrow y = $
3. c) If $x = 5 \rightarrow y = $
4. d) If $x = 10 \rightarrow y = $

Lucía is monitoring her cell phone battery during the morning. When she woke up at $08:00$, it was at $100\%$. For every hour that passes, the battery decreases by $8\%$.

1. a) Write the algebraic expression of the function relating the elapsed hours ($x$) to the remaining battery percentage ($y$).
2. b) What battery percentage will Lucía have at $12:00$ in the morning (after $4$ hours of use)?

An electric scooter has a battery that discharges at a constant rate. When leaving home, the battery is at $100\%$. For every kilometer traveled, the battery decreases by $5\%$.

1. a) Identify the independent variable and the dependent variable, stating the units in which each is measured.
2. b) Write the algebraic expression (formula) of the function relating the kilometers traveled ($x$) to the remaining battery percentage ($y$).
3. c) Is it possible for the value of $x$ to be $25$? Explain whether that value belongs to the domain of the function in this real-world context.

Complete the following table of values for the function $y = x^2 - 3$:

Once completed, answer:

1. a) Are there different values of $x$ that have the same image ($y$)? Does this prevent it from being a function?
2. b) Plot the points obtained on a set of coordinate axes. What shape does the graph appear to have?

You can do it on paper and send a photo with your answer.

In a strategy video game, the position of resources is given using coordinates on a Cartesian plane. Look at the map and write down the coordinates of the three treasure chests shown:

Write down the coordinates of each chest in the form $(x, y)$.

Observe carefully the following three graphs drawn on the Cartesian plane. Use the vertical line test (imagine vertical lines sweeping across each graph and count at how many points they cross it) to identify which of them represent a function, and justify your answer in detail in each case.

In a video game tournament, we want to establish a relation between players and the characters they choose. Analyse the following two situations and determine which one defines a function:

• Situation 1: Each player is assigned the name of the only character they have used throughout the entire tournament.

• Situation 2: Each character is assigned the names of all the players who have chosen it in any game.

Draw a mapping diagram (Venn diagram) for Situation 1 using these data: Players $\{Ainhoa, Marc, Leo\}$ and Characters $\{Guerrero, Maga\}$. Ainhoa and Marc chose the Warrior, and Leo chose the Mage.

You can do it on paper and send a photo with your answer.

Analyse carefully the following graphical representation showing the relationship between two quantities. Determine, reasoning your answer based on the definition of a function, whether this graph represents a function or a non-functional relationship. Also state what the value (or values) of the dependent variable $y$ would be when the independent variable $x$ is equal to $2$.

A relation between two sets of numbers is defined as follows: "To each integer $x$ corresponds its square root".

1. a) Is this relation a function if the initial set (domain) consists of all integers $\mathbb{Z}$? Justify your answer by analysing what happens with negative numbers.
2. b) What if we restrict the domain to the numbers $\{0, 1, 4, 9\}$? Would it be a function if each number can have two associated values (positive and negative)? Reason your answer based on the uniqueness of the image.

Given the following table of values collected during a physics experiment, it is suspected that one of the values is incorrect and breaks the condition for a function.

1. a) Identify which value prevents this table from representing a function. Justify your choice.
2. b) If we remove the incorrect value, would the independent variable be discrete or continuous?
3. c) Write a possible equation $y = f(x)$ relating position to time if the speed were constant at $12\ m/s$.

State whether the following situations represent a continuous function or a discrete function:

1. a) The number of people entering a cinema depending on the film being shown.
2. b) The altitude of a hot-air balloon as the minutes pass after takeoff.
3. c) The price of a bag of oranges depending on its mass in kilograms.
4. d) An influencer's number of Instagram followers day after day.

The following graph shows the distance of a delivery drone from its logistics base during a 10-minute round trip.

From the graph, state:

1. a) The domain and the range of the function.
2. b) How long was the drone stopped delivering the package? At what distance from the base was it at that time?
3. c) In which time interval was the drone returning to the base?

We are filling a cylindrical water tank using a faucet that pours a constant flow rate of $4$ liters per minute. The tank has a total capacity of $20$ liters.

1. a) State whether the function relating the elapsed time to the volume of water is continuous or discrete. Explain why.
2. b) What is the domain of this function, taking into account that the tank stops when full?
3. c) Draw a sketch of the graph representing this situation.

Given the function defined by the following graph, which represents the cost of a monthly subscription to a music platform as a function of the number of months subscribed:

1. a) Is it correct to join the points with a continuous line? Justify whether the independent variable is discrete or continuous.
2. b) If the point $(1, 10)$ belongs to the function, what does this datum mean in the context of the problem?
3. c) Determine the image of $x=3$ by observing the trend of the points.

A drone delivery company charges a base rate of $\text{RD\$}\ 300$ for the service. In addition, the cost for distance is divided into steps: the first $2\ km$ are free, but from then on, each additional kilometer or fraction of a kilometer costs $\text{RD\$}\ 120$ (that is, if it travels $2.1\ km$ it is charged as if it were $3\ km$).

1. a) Explain why this situation represents a discontinuous function.
2. b) Identify the independent variable and the dependent variable.
3. c) Draw a graph of the cost of the service for distances between $0$ and $5\ km$.

An athlete is training on a straight path. The following graph shows the distance from the starting point as a function of time.

1. a) Explain what is happening between minute $4$ and minute $6$.
2. b) What maximum speed did the athlete run at and in which interval did it happen?
3. c) Explain what is happening between $t=6$ and $t=8$.
4. d) Can we state that it is a continuous function? Why?

Consider the function that assigns each two-digit natural number the sum of its digits.

For example: $f(14) = 1 + 4 = 5$.

1. a) What is the domain of this function?
2. b) What is the range of the function? (Take into account the smallest and largest possible numbers).
3. c) Is it an increasing function? Justify your answer by comparing, for example, $f(19)$ and $f(20)$.

On a set of coordinate axes, draw the graph of a function that simultaneously satisfies the following four conditions:

1. Its domain is the interval $[-4, 6]$.

2. It has a point of discontinuity (a jump) at $x = 2$.

3. It is decreasing on the interval $[-4, 0]$ and increasing on the interval $[3, 6]$.

4. It has a relative maximum at the point $(-4, 5)$ and an absolute minimum at $(3, 1)$.

You can do it on paper and send a photo with your answer.