Characteristics of Functions

For a relationship 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 reason whether it represents a function or not using the "vertical line test".

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

Look at the graph of the following function (represented by the orange line segment) and write its domain and range using intervals.

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 interruptions, it is called discontinuous.

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

In a courier company, the cost of sending a parcel depends on its weight. However, the rate changes abruptly when exceeding certain weight brackets. Look at 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 range of this function according to the drawing.

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

A smart energy management system for a bioclimatic house regulates the heating according to the outside temperature. The function $f(x)$ represents the expense in dollars per hour in terms of the temperature $x$ in degrees Fahrenheit:

$$f(x) = \begin{cases} -0,5x + 8 & \text{if } x < 10 \\ 3 & \text{if } 10 \le x < 20 \\ 0,1x^2 - 4x + 7 & \text{if } x \ge 20 \end{cases}$$

Analyze the following characteristics of the function:

1. a) Determine the domain and the range of the function for the temperature interval $[-10, 30]$.
2. b) Study the continuity. Are there any points of discontinuity? If so, state what type they are and their meaning in the context of energy saving.
3. c) Find the intervals of monotonicity (increasing and decreasing).

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

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

The following graph shows the evolution of a smartphone's battery percentage throughout a day, from when the user wakes up at $08:00$ until they put it on charge at night.

Based on the graph, analyze the following characteristics:

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

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

3. Determine the maximum and minimum battery values in the observed interval.

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

Looking at the graph, answer:

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

2. State the approximate coordinates of the relative maxima and minima.

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

The maxima are the highest points in an area of the graph, and the minima are the lowest points.

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

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

Based on the previous graph, answer:

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? Justify your answer based on the monotonicity of the function.

A forest surveillance drone takes off to carry out an inspection. Its height $h$ in feet with respect to time $t$ in minutes is given by the attached 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 the relative and absolute maximums and minimums.
3. c) If the drone needs a minimum height of $10$ feet 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.

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

Analyze the graph and answer:

1. a) What is 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 intersect the $y$ axis (y-intercept) and what does it represent?

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

1. Calculate the points of intersection 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 with 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 that goes from $(0,0)$ to $(10,0)$. The third vertex of the triangle moves along the parabola $y = -x^2 + 10x$.

1. a) Find the analytical expression of 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 definition 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 height does the triangle have at that point?

Functions can present symmetry. A function is even if it is symmetrical with respect to the $y$ axis (like a mirror). It is odd if it is symmetrical with respect to the origin of coordinates $(0,0)$.

Determine what type of symmetry (even, odd or neither) the following two graphs have:

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 $Y$.

Analyze the symmetry of the function:

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

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

3. State the point of intersection with the x-axis ($X$ axis) and the point of intersection with the y-axis ($Y$ axis) for this structure.

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

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

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

Based on the visual information:

1. Explain with reasons why this function is periodic.

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

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

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

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 h if at 6:00 h 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 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 airplane takes off.

1. a) Determine the points of intersection of the function with the axes. What meaning does the point $(0,0)$ have in this context?
2. b) Using a table of values, study the trend of the function when the time $t$ becomes very large. Is there an asymptote?
3. c) Is the function even, odd or neither? Prove it analytically by calculating $R(-t)$.

A biologist studies the population of a species of insects in a laboratory. The population $P$ (in thousands) varies according to the temperature $T$ of the room (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 stop making physical sense?
2. b) Find the points of intersection with the x-axis. What do these values mean for the insect population?
3. c) Are there any intervals of the independent variable where the function is not continuous? Justify whether the biologist should avoid any specific temperature.

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

1. a) Study the range of the function. Can the rate be negative? Can it exceed the value $1$?
2. b) Analyse the symmetries of the function. If it is symmetrical, explain what this implies regarding positive and negative concentrations of the reactant.
3. c) Is it a bounded function? Observe its behavior when $x$ increases indefinitely and describe its horizontal asymptote.

A water tank is being filled by a pump that delivers a constant flow. At $10$ minutes, the pump is stopped for $5$ minutes to check a valve, and then filling resumes more slowly.

1. Sketch a graph representing the height of the water 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.

Two athletes, Ashley and Brandon, compete in a 10 mile race. The distance travelled by Ashley follows the function $d_A(t) = 0,4t$, while Brandon's is $d_B(t) = 0,01t^2$ (where $t$ is the time in minutes).

1. a) Plot both functions on the same axes for the time interval necessary until both reach the finish line.
2. b) At what exact moment does Brandon catch up with Ashley? State the coordinates of the point of intersection and interpret its meaning.
3. c) Study the growth of the difference function $g(t) = d_A(t) - d_B(t)$. At what moment is the distance between Ashley and Brandon maximum?