Characteristics of a Function

For a relation to be considered a function, each value of the independent variable ($x$) must correspond to, at most, a single value of the dependent variable ($y$).

Look at the following graph and explain whether it represents a function or not using the "vertical line test".

The domain of a function is the set of values taken by the variable $x$, while the range (or image) is the set of values taken by the variable $y$.

Look at the graph of the following function (represented by the orange segment, with both endpoints included) and write its domain and its range using intervals.

1. a) Write the domain of the function.

1. b) Write the range of the function.

A biologist is studying the population of an insect species in a laboratory. The population $P$ (in thousands) varies according to the room temperature $T$ (in $ºC$) following the function:

$$P(T) = \frac{T^2 - 16}{T - 5}$$

1. a) Calculate the domain of this function. For what temperature does the model cease to make physical sense?
2. b) Find the x-intercepts. What do these values mean for the insect population?
3. c) Are there intervals of the independent variable where the function is not continuous? Justify whether the biologist should avoid any specific temperature.

A smart energy management system for a bioclimatic home controls when the heating switches on depending on the outdoor temperature. The function $f(x)$ represents the cost in pesos per hour as a function of the temperature $x$ in degrees Celsius:

$$f(x) = \begin{cases} -0.5x + 8 & \text{if } x < 10 \\ 3 & \text{if } 10 \le x < 20 \\ 0.1x^2 - 4x + 7 & \text{if } x \ge 20 \end{cases}$$

You have its graph on the temperature interval $[-10, 30]$. Using it and the expression of $f$, analyze the following characteristics of the function:

1. a) Determine the domain and the range of the function for the temperature interval $[-10, 30]$.

1. b) Study the continuity. Are there any points of discontinuity? If so, state what type they are and what they mean in the context of energy saving.

1. c) Find the intervals of monotonicity (increasing and decreasing).

We say that a function is continuous if we can draw its graph without lifting the pencil from the paper. If it has jumps or breaks, it is called discontinuous.

The following graph shows the parking price at a shopping center based on the hours elapsed. State whether the function is continuous or discontinuous and, if discontinuous, indicate at which values of $x$ the jumps occur.

In a delivery company, the cost of sending a parcel depends on its weight. However, the price changes abruptly when exceeding certain weight bands. Observe the graph of the cost function:

Answer the following questions about continuity:

1. Is this function continuous? Justify your answer by describing what happens at the points $x = 2kg$ and $x = 5kg$ (represented by the jumps in the graph).

2. State the domain and the range of this function according to the diagram.

3. If a parcel weighs exactly $2kg$, what is its cost? Look at the solid and open dots to decide.

A function is increasing if as $x$ increases, $y$ increases. It is decreasing if as $x$ increases, $y$ decreases.

Marcos is on a hiking route. The following graph represents his altitude at any given time. Identify the time intervals ($x$ axis) in which Marcos is going up (increasing function) and those in which he is going down (decreasing function).

Maximums are the highest points in a section of the graph, and minimums are the lowest points.

In a video game, a character's energy varies according to the graph shown. State the $(x, y)$ coordinates of the points representing a relative maximum and a relative minimum in this adventure.

Marta uses an electric scooter to go to school. The following graph shows the remaining battery percentage as a function of the distance traveled in kilometers.

Analyze the graph and answer:

1. a) What are the domain and range of the function?
2. b) Is it an increasing or decreasing function? What does this mean in the context of the problem?
3. c) At what point does the graph cross the $y$ axis (y-intercept) and what does it represent?

The following graph shows the changes in battery percentage of a smartphone over a day, from when the user wakes up at $08:00$ until putting it on charge at night.

Based on the graph, analyse the following features:

1. State the time intervals in which the function is decreasing and explain what this means in this context.

2. Is there any interval where the function is constant? If so, state it and propose a real-life situation that explains why the battery level does not decrease during that time.

3. Determine the maximum value and the minimum value of the battery in the observed interval.

A YouTube channel analyses the number of "Likes" a video receives during the first $12$ hours after publication. The function is represented by the following graph:

Looking at the graph, answer the following:

1. Identify the intervals where the function is increasing and decreasing.

2. State the approximate coordinates of the relative maximums and minimums.

3. What is the absolute maximum of the function in this period? Explain the difference between this point and a relative maximum.

A water tank is being filled with a pump that pours water at a constant flow rate. After $10$ minutes, the pump stops for $5$ minutes to inspect a valve, and then filling resumes more slowly.

1. Sketch a graph representing the water height as a function of time.

2. Identify whether the resulting function is continuous or discontinuous.

3. State the time intervals in which the function is increasing and the interval in which it is constant.

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

The temperature of an industrial oven varies during a smelting process. It is switched on, rises to its working temperature, remains stable, and then cools down quickly when the door is opened.

Based on the graph above, answer the following:

1. What is the domain of the function (total time of the process)?

2. What is the absolute maximum temperature and in what time interval is it maintained?

3. At what moment is the oven door opened? Reason your answer based on the monotonicity of the function.

A forest surveillance drone takes off to carry out an inspection. Its height $h$ in meters with respect to time $t$ in minutes is given by the accompanying graph. The mission lasts exactly 60 minutes.

1. a) State the intervals where the function is constant and explain what the drone is doing at those times.
2. b) Identify all relative and absolute maximums and minimums.
3. c) If the drone needs a minimum height of $10$ meters to detect heat, in which time intervals is its mission effective?
4. d) Is it a periodic function? Give reasons for your answer based on the shape of the graph.

Given the function $f(x) = x^2 - 4x + 3$, carry out a study of its main features:

1. Find the intercepts with the $X$ axis (by solving the equation $f(x) = 0$) and with the $Y$ axis (by calculating $f(0)$).

2. Find the coordinates of the vertex, knowing that the x-coordinate is calculated using the formula $x = \frac{-b}{2a}$. Is it an absolute maximum or minimum?

3. State the interval where the function is decreasing.

Consider a triangle whose base is fixed on the segment of the Cartesian axes extending from $(0,0)$ to $(10,0)$. The third vertex of the triangle moves along the parabola $y = -x^2 + 10x$.

1. a) Find the algebraic expression for the function $A(x)$ that represents the area of the triangle in terms of the x-coordinate $x$ of the third vertex.
2. b) What is the domain of this function for the area to be a positive value?
3. c) Calculate the coordinates of the vertex of the area function and state whether it represents an absolute maximum or minimum. What is the height of the triangle at that point?

The noise level in decibels ($dB$) in the vicinity of an airport can be modelled by the function $R(t) = \frac{80t}{t^2 + 4}$, where $t$ is the time in minutes since an aircraft takes off.

1. a) Determine the axis intercepts of the function. What is the meaning of the point $(0,0)$ in this context?
2. b) Using a table of values, study the trend of the function as the time $t$ becomes very large. Is there an asymptote?
3. c) Is the function even, odd or neither? Prove it algebraically by calculating $R(-t)$.

Two athletes, Alba and Bruno, compete in a 16 km race. The distance covered by Alba follows the function $d_A(t) = 0.4t$, while Bruno's is $d_B(t) = 0.01t^2$ (where $t$ is the time in minutes and distances are measured in km).

1. a) Graph both functions on the same axes for the time interval needed until both reach the finish line.

1. b) At what exact moment does Bruno catch up with Alba? Give the coordinates of the intersection point and interpret its meaning.

1. c) Study the growth of the difference function $g(t) = d_A(t) - d_B(t)$. At what moment is the distance between Alba and Bruno greatest?

A function is periodic if its graph repeats exactly the same over a certain interval, called the period.

The signal from a heart rate monitor behaves in a repetitive way. Look at the graph and determine the value of the period (the distance on the $x$ axis needed to complete one full cycle).

Functions can display symmetry. A function is even if it is symmetrical about the $y$ axis (like a mirror). It is odd if it is symmetrical about the origin $(0,0)$.

Determine what type of symmetry (even, odd, or none) the following two graphs have:

A pressure sensor on an industrial machine records a repetitive cycle every few seconds. The pressure $P$ (in bars) depends on the time $t$ (in seconds) according to the following graph:

Based on the visual information:

1. Explain with reasoning why this function is periodic.

2. If the maximum pressure peaks occur at $t = 2$ s, $t = 10$ s and $t = 18$ s, what is the value of the period $T$?

3. What pressure will the sensor record at the instant $t = 82$ s?

An architect designs a decorative arch for a museum entrance whose shape follows a symmetrical function. The axis of symmetry coincides with the y-axis $Y$.

Analyse the symmetry of the function:

1. Explain the difference between an even function and an odd function. What type of function is this?

2. If we know that the point $(-3, 0)$ lies on the graph, what other point can we be sure lies on it by symmetry?

3. State the x-intercept ($X$ axis) and the y-intercept ($Y$ axis) for this structure.

A light sensor is being designed for an automated greenhouse. The light intensity $I$ (in lux) follows a pattern that repeats every 24 hours. We know that at 00:00 (time $t=0$) the intensity is $0$ lux, and it reaches its maximum of $1200$ lux at 12:00.

1. a) If the function is symmetrical with respect to the line $t=12$ within each 24-hour cycle, what intensity will there be at 18:00 if at 06:00 it is $450$ lux?
2. b) Knowing that the function is periodic with period $T=24$, calculate the value of the light intensity for the times $t=30$, $t=54$ and $t=108$ hours.
3. c) Could this be an odd function? Give reasons for your answer by analysing the range of the intensity (which cannot be negative).

The rate of a chemical reaction depends on the concentration of a reactant $x$. The function that describes this relationship is $v(x) = \frac{x^2}{x^2 + 1}$. You can see its graph in the figure.

1. a) Study the range of the function. Can the rate be negative? Can it exceed the value $1$?

1. b) Analyze the symmetries of the function. If it is symmetric, explain what this implies for positive and negative concentrations of the reactant.

1. c) Is it a bounded function? Observe its behavior as $x$ increases without limit and describe its horizontal asymptote.