Exercises: How functions are presented

Observe the following graphs and state which of them represent a mathematical function and which do not. Justify your answer using the vertical line test.

Express through an algebraic formula the following relationship described verbally: "The cost $C$ of buying $x$ kilograms of apples if each kilo costs $1,80$ € and the plastic bag to carry them costs $0,10$ €."

Determine the algebraic expression of the linear function that relates the variables $x$ and $y$ given in the following table of values:

• $x$: $1$, $2$, $3$, $4$, $5$

• $y$: $5$, $7$, $9$, $11$, $13$

Given the function defined by the formula $f(x) = \frac{x^2 - 1}{2}$, construct a table of values for $x \in \{-3, -1, 0, 1, 3\}$.

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

From the following graph, construct a table of values for the coordinates $x \in \{-4, -2, 0, 2, 4\}$.

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

Observe the following tables of values and determine which one represents a function and which one does not. Justify your answer by stating the condition a relationship must satisfy to be a function.

Table A:

• $x$: $-2$, $0$, $2$, $2$, $4$

• $y$: $4$, $0$, $3$, $5$, $16$

Table B:

• $x$: $-2$, $-1$, $0$, $1$, $2$

• $y$: $4$, $1$, $0$, $1$, $4$

A paid car park charges a base fee of $2$ € to enter and $1,50$ € for each hour of stay. Create a table of values showing the total cost for stays of $0$, $1$, $2$, $3$, and $4$ hours.

Analyse the following verbal descriptions and decide which ones define a mathematical function (where each initial element corresponds to a unique final element):

1. a) Each person is associated with their ID number.
2. b) Each shoe size is associated with a person who wears it.
3. c) Each real number is associated with its square.
4. d) Each positive real number is associated with its square roots.

An urban mobility company in a Spanish city offers a shared electric scooter rental service. The fare consists of a fixed unlock fee of $1$ € plus $0,25$ € for each minute of the ride.

1. Write the algebraic expression that gives the total cost of the trip, $C(t)$ in euros, as a function of the usage time $t$ expressed in minutes.

2. Create a table of values showing the cost for trips of $0$, $4$, $10$, $15$, and $20$ minutes.

3. For safety and battery life reasons, the app automatically locks any ride upon reaching $45$ minutes of continuous duration. Taking this into account, define with reasoning the contextual domain of the function $C(t)$.

A courier service applies a tiered rate for peninsular shipments according to the package's weight in kilograms ($x$). The rate in euros is given by the function:

$P(x) = \begin{cases} 4,50 & \text{if } 0 < x \le 2 \\ 4,50 + 1,20 \cdot (x - 2) & \text{if } x > 2 \end{cases}$

Lucía needs to send a package that weighs exactly $1,5$ kg. To calculate the price, she directly substitutes $x = 1,5$ into the second formula:

$P(1,5) = 4,50 + 1,20 \cdot (1,5 - 2) = 4,50 + 1,20 \cdot (-0,5) = 4,50 - 0,60 = 3,90 \text{ €}$

Lucía concludes that, since her package weighs $1,5$ kg, it is cheaper for her than the fixed rate of $4,50$ € for small packages.

Explain what conceptual mistake Lucía made when evaluating the piecewise function, reason why her conclusion makes no physical sense in the context of the problem, and calculate the correct price she must pay.

Two phone companies offer different monthly tariffs. Company Alpha charges a fixed fee of $5$ € plus $0,10$ € for each minute of call. Company Beta has no fixed fee, but charges $0,15$ € per minute.

1. Write the algebraic expression for the monthly cost for each company as a function of the minutes used $x$.

2. Calculate how much you would pay with each company if you use $120$ minutes in a month.

3. Find algebraically from how many minutes Company Alpha becomes cheaper.

A piece of wire $30$ cm long is available to form a rectangle.

1. Express the area of the rectangle $A(x)$ as a function of the measurement of its base $x$.

2. What is the logical domain of this function in the context of the problem? That is, what real values can the base $x$ take for the rectangle to exist?

The following graph shows the variation in outdoor temperature at a weather station over a full day ($24$ hours). From the graph, answer:

1. What are the maximum and minimum temperatures reached, and at what time do they occur?

2. State the time intervals in which the temperature increases and in which it decreases.

Match each of the following algebraic formulas with its corresponding table of values:

Formulas:

1. $y = 2x - 3$

2. $y = x^2 + 1$

3. $y = \frac{12}{x}$

Tables:

• Table A: $(-2, 5)$, $(0, 1)$, $(2, 5)$, $(3, 10)$

• Table B: $(-2, -6)$, $(2, 6)$, $(3, 4)$, $(4, 3)$

• Table C: $(-1, -5)$, $(0, -3)$, $(2, 1)$, $(4, 5)$

In a physics experiment, a stone is dropped from a bridge $45$ metres above a river. The height $h$ of the stone above the water, measured in metres, as a function of time $t$ in seconds, is given by the expression $h(t) = 45 - 5t^2$.

1. Construct a table of values for $t = 0$, $t = 1$, $t = 2$, and $t = 3$.

2. At what exact moment in time does the stone hit the water?

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

Identify which of the following equations relate $x$ and $y$ such that $y$ is a function of $x$. Justify your answer by rearranging for the variable $y$ where possible.

1. a) $3x + 2y = 6$
2. b) $x^2 + y^2 = 25$
3. c) $y = x^3 - 2$

A taxi service charges a standing charge of $2,50$ € and a surcharge of $1,20$ € for each kilometre travelled.

1. Write the analytical expression of the function that gives the cost of the trip as a function of the kilometres travelled.

2. If a customer paid $14,50$ € upon reaching their destination, how many kilometres did the taxi travel?

Translate the following situation into a piecewise algebraic expression: "A youth travel pass costs $20$ € per month and allows up to $30$ journeys at no extra cost. From journey number $31$ onwards, $0,50$ € is charged for each extra journey made." Let $x$ be the total number of journeys made in the month and $C(x)$ be the total monthly cost.

According to data from the Spanish State Meteorological Agency (AEMET), on a spring day in Sierra Nevada the temperature at sea level ($0$ metres of altitude) is $18$ ºC. As you climb the mountain slope, the temperature drops at a constant rate of $0,6$ ºC for every $100$ metres of elevation gain.

1. Obtain the formula of the function $T(h)$ that relates the temperature in degrees Celsius to the altitude $h$ measured in metres.

2. If the highest accessible summit on the route is at Veleta peak, at $3396$ metres of altitude, determine the contextual domain of the function, justifying the endpoints of the interval.

3. Calculate from what altitude the temperature will be below freezing point ($0$ ºC) and interpret whether hikers will encounter snow or ice before reaching the summit.

The coordinator of a municipal sports centre wants to analyse gym occupancy during a Saturday, from opening at 8:00 to closing at 22:00. To present the results to the management team, they can choose among three different ways of displaying the same information:

• Form 1 (Table): A list with the exact number of people recorded by the entrance turnstile every hour on the hour ($8$ h, $9$ h, $10$ h, etc.).

• Form 2 (Graph): A continuous time curve generated by the capacity sensor in real time throughout the $14$ hours.

• Form 3 (Formula): An approximate degree 3 polynomial algebraic expression, $A(t) = -0,2 \cdot t^3 + 4,1 \cdot t^2 - 18 \cdot t + 25$, where $t$ is the number of hours elapsed since 8:00.

Answer the following questions with justification:

1. a) If the management needs to identify at a glance the peak rush hours of highest occupancy and the intervals during which the gym was below $30$ % capacity, which of the three forms is the most effective and why?
2. b) If there is a need to audit compliance with the legal capacity limit at 12:00 sharp for a technical inspection, which is the most convenient form and what drawback would using the formula present?

Look at the graph of the function $f(x)$, which consists of three pieces: part of a hyperbola, a line segment, and part of an exponential curve.

1. a) Determine, by looking at the grid (1 unit = 1 square), the domain and range of the function.
2. b) Identify the equation of the vertical line marked in red (asymptote of the hyperbola to which the first piece belongs) and the points of discontinuity.
3. c) Knowing that the first piece is of the form $y = \frac{k}{x+2}$, the second is linear, and the third is of the form $y = a^{x-3}$, determine the complete analytical expression of the piecewise function $f(x)$.

A square sheet of cardboard with side length $40$ cm is available. An open box is to be constructed by cutting a square of side $x$ from each of the four corners and folding the resulting flaps upwards.

1. Express the volume of the box, $V(x)$, as a function of the side $x$ of the cut-out square.

2. Determine the domain of definition of this function in the geometric context of the problem.

The rates for a public car park indicate that $2$ € will be charged for the first hour or fraction thereof, and $1,50$ € for each additional hour or fraction thereof.

1. Construct a table of values showing the cost of parking for times $t \in \{0,5; 1; 1,5; 2; 2,5\}$ hours.

2. Explain why the graphical representation of this situation corresponds to a step function (a piecewise function that is constant on intervals).

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

From the graph of the following piecewise function, find the algebraic expression (formula) corresponding to each of the three straight sections that make it up.

A theatre has $500$ seats. It has been verified that, if the ticket price is $20$ €, the theatre sells out completely. In addition, a market study estimates that for each $5$ € increase in ticket price, $40$ fewer seats are sold.

Express the total revenue of the theatre, $I(x)$, as a function of the number of $5$ € increases applied to the original price (that is, $x$ represents the number of times the price is raised by $5$ €).

Observe the following sequence of figures made of dots. Determine the algebraic expression of the function $N(s)$ relating the figure number (step $s$) to the total number of dots that form it. Use the formula obtained to calculate how many dots Figure 40 will have.

Find the algebraic expression of the parabola shown in the graph knowing that it is of the form $y = ax^2 + bx + c$. Use the highlighted points to set up and solve the necessary system of equations.

A cyclist $A$ leaves a city heading east at a constant speed of $20$ km/h. Exactly one hour later, a second cyclist $B$ leaves the same city in the same direction pursuing the first at a constant speed of $25$ km/h. Express the distance $d(t)$ separating both cyclists as a function of the time $t$ (in hours) elapsed since cyclist $A$ set off.

A tank with a capacity of $500$ litres initially contains $100$ litres of water. A tap is turned on, pouring water at a rate of $25$ litres per minute for $10$ minutes. Next, the tap is turned off and the tank remains undisturbed for $5$ minutes. Finally, a drain is opened, emptying the water at a rate of $50$ litres per minute until the tank is completely empty. Write the piecewise analytical expression of the volume function $V(t)$ indicating the amount of water in the tank at time $t$ (in minutes).

In a Spanish provincial capital, the public water utility bills bi-monthly domestic water consumption in tiers to reward conservation, applying the following variable rate to the cubic metres ($x$) consumed:

• Tier 1 (basic consumption, up to $20 \text{ m}^3$): $0,50$ € per $\text{m}^3$.

• Tier 2 (average consumption, excess over $20 \text{ m}^3$ and up to $40 \text{ m}^3$): the first $20 \text{ m}^3$ are charged at the Tier 1 rate, and cubic metres exceeding $20$ are charged at $1,10$ € each.

• Tier 3 (high consumption, over $40 \text{ m}^3$): the first $40 \text{ m}^3$ according to the previous tiers, and the excess over $40 \text{ m}^3$ at $2,20$ € per $\text{m}^3$.

A household has recorded a bi-monthly consumption of $35 \text{ m}^3$.

1. Without calculating the exact bill yet, make an estimate of the cost: will the total amount be closer to $17$ €, $27$ €, or $45$ €? Justify how you rule out the other two options relying on the tier prices.

2. Write the complete analytical expression of the piecewise function $C(x)$ that calculates the cost of water for any consumption $x$ in the interval $[0, 40]$.

3. Calculate the exact amount for the $35 \text{ m}^3$ consumed, check the validity of your initial estimate, and calculate the actual average price this family pays for each cubic metre consumed.

A tourist municipality is going to launch a supervised park-and-ride public car park and needs to set hourly rates to prevent vehicles from being left abandoned for several days. The following requirements have been established for the total cost function $P(t)$ in euros as a function of the parking time $t$ in hours:

1. The first two hours ($0 \le t \le 2$) are free ($0$ €) to encourage local trade.

2. Between $2$ and $6$ hours ($2 < t \le 6$), the cost increases linearly at a rate of $1,80$ € for each additional hour beyond the second.

3. From $6$ hours up to the maximum daily limit of $24$ hours ($6 < t \le 24$), the price per additional hour increases to penalise long stays. The function must be continuous (with no jump at $t = 6$), so that the cost starts at the value accumulated up to that point and reaches a final amount of exactly $61,20$ € if the car stays for the full $24$ hours.

With these data:

1. a) Determine the analytical expression of the piecewise function $P(t)$, detailing the formula and the interval of validity for each of its three pieces.
2. b) If a user has paid a total of $16,20$ €, find out with justification, using the equation of the corresponding piece, exactly how many hours their vehicle has remained in the car park.