Concept of a Function

In an app store, each premium game has a fixed price of $\$2,50$. We want to analyse the relationship between the number of games bought and the total cost of the purchase.

1. a) Identify which is the independent variable and which is the dependent variable.
2. b) Is this relationship a function? Give reasons for your answer based on the definition of a function.
3. c) Write the algebraic expression (formula) that relates both quantities, calling $x$ the number of games and $y$ the total cost.

Observe the following relationships between quantities and determine, in each case, whether it is a function or not:

1. a) Each mobile phone number is associated with its owner's name in the contacts.
2. b) Each student in your class is associated with their height measured in inches.
3. c) Each barcode in a supermarket is associated with the price of the corresponding product.
4. d) Each person is associated with the eye color of their siblings.

In a middle school, different relationships between quantities are being analysed. Determine with reasons which of the following situations correspond to a function and, if so, identify the independent variable and the dependent variable:

1. a) Each student's ID number is associated with their current age.
2. b) Each maximum temperature reached in the playground during a day is associated with the times at which that temperature was recorded.
3. c) The amount of gigabytes ($GB$) consumed on a mobile plan is associated with the monthly price of the bill.
4. d) Each natural number is associated with its divisors.

A hiker climbs a mountain following a path. The function that relates their altitude above sea level, $H$ (in feet), to the horizontal distance travelled, $d$ (in miles), is $H(d) = 400 + 150d$.

1. a) At what altitude is the starting point of the route ($d = 0$)?
2. b) If the hiker has travelled $3$ miles horizontally, at what altitude are they?
3. c) Identify the variables of the function and explain their physical meaning.

An electric scooter rental company charges a fixed fee of $\$1,20$ to unlock the vehicle and a variable fee of $\$0,15$ for each minute of use.

1. a) Identify the two variables involved and classify them as dependent and independent.
2. b) Write the analytical expression or formula that allows calculating the total cost of the rental, $y$, in terms of the time of use in minutes, $x$.
3. c) Calculate how much a student will pay who uses the scooter for $22$ minutes to go from their house to the school.

The area of a rectangle whose base is twice its height can be expressed as a function.

1. a) If we call $x$ the measure of the height of the rectangle, write the analytical expression of the area function $A(x)$.
2. b) Which is the independent variable and which is the dependent one?
3. c) Calculate the area when the height is $4,5$ $cm$.
4. d) If the area of the rectangle is $50$ $cm^2$, what is its height?

Given the function defined by the algebraic expression $f(x) = 3x - 2$, complete the following table of values in your exercise book:

$x$ (Input)	$-2$	$-1$	$0$	$1$	$2$
$f(x)$ (Output)

Once completed, state what the image of $x = -1$ is.

Given the function $f(x) = \frac{x^2 - 4}{2}$, carry out the following tasks:

1. a) Calculate the images of the values $x = 0$, $x = 2$ and $x = -4$.
2. b) Determine the value of $x$ (pre-image) whose image is $f(x) = 6$.
3. c) Does the point $P(-1, -1.5)$ belong to the graph of the function? Justify your answer by performing the necessary calculations.

A water tank has the shape of an inverted cone. It is being filled with water at a constant rate. The relationship between the volume of water ($V$) and the height the water reaches ($h$) is given by the function:

$$V(h) = \frac{1}{3} \cdot \pi \cdot (0,5h)^2 \cdot h$$

1. a) Simplify the expression of the function $V(h)$.
2. b) Is it a linear function? Justify your answer by observing the degree of the variable $h$.
3. c) Calculate the pre-image of $V = \frac{2\pi}{3}$, that is, the height $h$ needed to have that volume.
4. d) If the tank has a total height of $4 metros$, what is the domain and the range of the function in this situation?

The price of a monthly subscription to a streaming service is $\$9$, but the first month has an offer and only costs $\$3$.

1. a) Create a table relating the number of months subscribed (from $1$ to $4$ months) to the total accumulated expense.
2. b) Plot these four points on a Cartesian coordinate system.
3. c) Does it make sense to join the points with a continuous line? Explain why.

The following table shows the amount of water (in gallons) remaining in a portable swimming pool while it is being emptied by an extraction pump:

Time (minutes)	0	5	10	15	20
Water (gallons)	4500	3600	2700	1800	900

1. a) Is it a functional relationship? Explain why.
2. b) Plot the points on a coordinate axis system choosing a suitable scale.
3. c) If the trend remains constant, how long will it take for the pool to empty completely?
4. d) What volume of water will there be $12$ minutes after starting to empty it?

Observe the following table of values that relates two quantities, $A$ and $B$:

Quantity A ($x$)	-2	-1	0	1	2
Quantity B ($y$)	5	2	1	2	5

1. a) Find an analytical expression (formula) that relates $y$ to $x$.
2. b) State whether it is a function and justify why.
3. c) Determine the points of intersection with the axes.
4. d) If we expand the table with the value $x = 1$ and assign it $y = 3$, would it still be a function? Give reasons for your answer.

Olivia is monitoring the battery consumption of her smartphone while playing a video game. The following graph shows the percentage of battery remaining according to the time she has been playing (in minutes):

From the graph, answer:

1. a) What percentage of battery did she have when she started playing (time = $0$)?
2. b) How much battery does she have left after $20$ minutes of playing?
3. c) If the battery reaches $0\%$, how many minutes has she been playing in total?

Observe the points plotted on the following Cartesian plane, which show the temperature of a greenhouse during the early hours of the morning:

State the coordinates $(x, y)$ of the four points marked on the graph and explain what each of them means in the context of the problem. (Example: At time $1$, the temperature is... ).

A hiker records the percentage of battery remaining on their smartphone during a mountain walk. The resulting graph is as follows:

Looking at the graph, answer:

1. a) What are the independent and dependent variables? State the units of measurement.
2. b) What was the battery percentage at the start of the walk?
3. c) Is it an increasing or decreasing function? What physical meaning does this have in the context of the problem?
4. d) Estimate how much time has passed when the phone has exactly $50$% of charge left.

For a graph to represent a function, any vertical line we draw must intersect the graph at a single point. Use this "vertical line test" to decide which of the following graphs is NOT a function:

Justify your choice by explaining what happens in the chosen graph when drawing a vertical line.

Observe the following graphs plotted on the Cartesian plane. Using the vertical line test, state with reasons which of them represents a function and which does not.

Analyze the following graphs and determine whether they represent a function or not. Justify your answer using the concept of the relationship between independent and dependent variables.

1. a) $x^2 + y^2 = 25$ (A circle of radius 5 centered at the origin).
2. b) $y = \sqrt{x}$ (Considering only the positive root).
3. c) A vertical line on the Cartesian plane passing through $x = 4$.
4. d) A horizontal parabola whose equation is $x = y^2$.

Consider the relationship that associates each integer $x$ with its modulus plus three units, that is, $y = |x| + 3$.

1. a) Create a table of values for the integers between $-3$ and $3$.
2. b) Plot the points obtained on a Cartesian coordinate system.
3. c) Does it make sense to join the points with a continuous line if the statement says that $x$ is an integer? Explain why.
4. d) State what the range (set of values taken by the dependent variable) would be based on your table and graph.

A video game programmer is designing the reward system for a new level. The amount of "Star Credits" ($C$) a player receives depends on the time ($t$) in minutes it takes to complete a mission, according to the relationship:

$$C(t) = \frac{600}{t-5}$$

1. a) State with reasons whether this relationship is a function for any value of $t$ greater than zero.
2. b) If the minimum physical time to complete the mission is 6 minutes and the maximum allowed by the server is 60 minutes, determine the domain and the range of this function in this real-life context.
3. c) What happens to the reward as the player approaches the time of 5 minutes? Would it make sense for the domain to include the value $t = 5$?

We want to build a rectangular wooden frame for a mirror. We have a wooden batten of $4$ feet in total length to make the entire outline.

1. a) If we call $x$ the length of the base of the frame (in feet), express the function $A(x)$ that calculates the area enclosed by the frame.
2. b) What is the domain of this function taking into account the physical restrictions of the problem?
3. c) Calculate the area when the base measures $0,5 m$ and when it measures $1,5 m$.
4. d) For what value of $x$ is the maximum area obtained? What geometric shape does the frame have in that case?

Consider the relationship that associates each real number $x$ with its distance to the nearest integer. For example, if $x = 2,3$, its distance to the nearest integer (2) is $0,3$. If $x = 2,8$, its distance to the nearest integer (3) is $0,2$.

1. a) Represent this relationship graphically in the interval $x \in [0, 3]$.
2. b) Explain why this relationship is a function.
3. c) Determine the range (image) of the function.
4. d) Is it a periodic function? If so, state its period.

A drone courier company charges for its services as follows: a fixed cost of $\$5$ for administration and, in addition, $\$2$ for each pound of weight. However, if the package exceeds $10 kg$, the cost per additional pound (only the pounds over 10) is halved.

1. a) Express the function $P(x)$ that represents the total price of the delivery in terms of the pounds $x$ of the package. Note: the function must have two expressions (a piecewise function).
2. b) Calculate $P(8)$ and $P(15)$.
3. c) Plot the graph of the function.
4. d) If a customer has paid $\$31$, how many pounds did their package weigh?

A mobile phone company offers a plan for teenagers called "Gamer Pro". The monthly cost ($C$) in dollars depends on the Gigabytes ($g$) consumed:

• The first $20 GB$ are free (included in the fixed fee of $\$15$).

• From $20 GB$ onwards, each extra Gigabyte costs $\$1,5$, but only up to a maximum of $50 GB$ of total consumption.

• If more than $50 GB$ are consumed, the speed is reduced and nothing more is charged.

1. a) Plot the graph of the function $C(g)$ for a consumption between $0$ and $80 GB$.
2. b) Define the function using a piecewise analytical expression.
3. c) State the intervals where the function is constant and where it is increasing.
4. d) What is the domain and the range of this function?