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.

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

Analyze the graph and answer:

1. a) What are the domain and the range of the function?

1. b) Is it an increasing or a decreasing function? What does this mean in the context of the problem?

1. c) At what point does the graph cross the $y$-axis (y-intercept) and what does it represent?

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 centre 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 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:

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

Cian is on a hiking route. The following graph represents his altitude at any given time. Identify the time intervals ($x$ axis) in which Cian 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.

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.

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.

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?