Funciones y sus Propiedades

Definition and Fundamental Concepts of Functions

A function is a relationship between two variables, typically denoted as $x$ and $y$. The independent variable, usually represented by $x$, is the input value that can be chosen freely within the allowed domain. The dependent variable, represented by $y$ or $f(x)$, depends on the value of $x$. For instance, when expressing the relationship between grams and milligrams, we can define a function where $x$ represents grams and $y$ represents milligrams. Given the conversion factor that $1\,g = 1000\,mg$, the function is expressed as $y = 1000 \times x$. To represent this graphically, one can construct a table of values: if $x = 1$, then $y = 1000$; if $x = 1.2$, then $y = 1200$; if $x = 0.5$, then $y = 500$. These points are then plotted on a Cartesian coordinate system where the horizontal axis represents the input ($x$) and the vertical axis represents the output ($y$).

Domain and Image of a Function

The domain ($Dom$) of a function consists of all the possible values that the independent variable $x$ can take such that the function is defined. Conversely, the image or range ($Im$ or $Recorrido$) consists of all the values that the dependent variable $y$ takes as a result of the inputs. For a linear function like $f(x) = 3x - 2$, the domain is the set of all real numbers, denoted as $Dom(f) = \mathbb{R}$. For a square root function such as $f(x) = \sqrt{x}$, the domain is restricted to non-negative numbers because the square root of a negative number is not a real number; thus, $Dom(f) = [0, +\infty)$. Regarding rational functions, the domain excludes values that make the denominator zero. For example, if $f(x) = \frac{1}{x}$, then $Dom(f) = \mathbb{R} - \{0\}$, and for $f(x) = \frac{1}{x-2}$, the domain is $Dom(f) = \mathbb{R} - \{2\}$. When analyzing graphs, the domain is observed along the $x$-axis, while the image is observed along the $y$-axis.

Practical Applications and Word Problems

Functions are used to model real-world scenarios, such as the entry of spectators into a venue. In a scenario where a cinema or theater opens at 19:00 with a maximum capacity of 250 people, and a certain number of guests enter over time, the relationship can be graphed with time on the $x$-axis and the number of spectators on the $y$-axis. If a function is defined as $y = 5x$, where $x$ is the number of minutes after 19:00, then after 25 minutes, there would be $5 \times 25 = 125$ spectators.

Another application involves geometry and perimeter. If Lidia wants to buy a rectangular fence where the length is related to the width, and the total perimeter or cable available is fixed, the area or length can be expressed as a function of the width $x$. For instance, if the total length available is $24\,m$ and the relationship for the perimeter is $2x + 2y = 24$, the dependent variable can be isolated to show how the dimensions vary relative to each other.

In a catering context, costs can be calculated per diner based on various menu items. If a meal cost includes fixed prices for different items (e.g., 30 for one item, 40 for another, 20 for a third, and 25 for a fourth), the total cost ($y$) becomes a function of the number of guests ($x$). If the sum of the components per person is $115$, the cost function is $y = 115x$. For example, for 2Guests, $y = 230$; for 4 guests, $y = 460$; and for 5 guests, $y = 575$. This can be summarized in a table and extended to higher numbers of diners, such as 6 guests costing $690$ or 10 guests costing $1150$.

Continuity and Cut Points

A function is considered continuous if its graph can be drawn without lifting the pencil from the paper. Points where the function is interrupted are called points of discontinuity. Cut points (or intercepts) are the intersections of the graph with the axes. To find the cut point with the $y$-axis, one sets $x = 0$ and solves for $y$. To find the cut points with the $x$-axis, one sets $y = 0$ and solves the resulting equation for $x$. For a quadratic function like $y = x^2 - 9$, the $y$-intercept is at $(0, -9)$ because $0^2 - 9 = -9$. The $x$-intercepts are found by solving $x^2 - 9 = 0$, resulting in $x = 3$ and $x = -3$, or the points $(3, 0)$ and $(-3, 0)$.

Complex functions require the quadratic formula to find cut points. For $y = x^2 + x - 6$, the $y$-intercept is $(0, -6)$. Setting the equation to zero: $x^2 + x - 6 = 0$. Using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we get $x = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-6)}}{2 \times 1} = \frac{-1 \pm \sqrt{25}}{2}$, yielding solutions $x = 2$ and $x = -3$. Thus, the cut points are $(2, 0)$ and $(-3, 0)$.

Monotonicity: Increasing and Decreasing Intervals

Monotonicity refers to the intervals where a function is increasing or decreasing. A function is increasing on an interval if, as $x$ increases, $y$ also increases. It is decreasing if, as $x$ increases, $y$ decreases. On a graph, these intervals are always specified using the $x$-values. Additionally, relative extrema are identified: a maximum is a point where the function changes from increasing to decreasing, and a minimum is a point where it changes from decreasing to increasing. For example, if a function represents noise levels in decibels ($dB$) over time, the intervals of increase might correspond to busier hours, while peaks represent maximum noise levels (e.g., $90\,dB$) at specific times.

Periodicity and Symmetry

A function is periodic if its values repeat at regular intervals called the period ($T$). This is expressed as $f(x) = f(x + T)$. Common examples include oscillating waves or social patterns that repeat daily or weekly.

Symmetry describes how a function behaves when the input is negated. There are two main types of symmetry:

1. Even Symmetry (Par): The function is symmetric with respect to the $y$-axis. This occurs if $f(x) = f(-x)$. An example is $f(x) = x^2$, since $(-x)^2 = x^2$.
2. Odd Symmetry (Impar): The function is symmetric with respect to the origin. This occurs if $f(-x) = -f(x)$. An example is $f(x) = x^3$, since $(-x)^3 = -x^3$.

To test for symmetry analytically, one substitutes $-x$ into the function and simplifies. If $f(x) = x^2 + 3$, then $f(-x) = (-x)^2 + 3 = x^2 + 3$, which is equal to $f(x)$, confirming even symmetry. If the resulting expression is neither the original function nor its exact negative, the function is asymmetric (neither even nor odd).

Summary of Domain Rules for Exams

When preparing for exams, the domain is often determined by the type of function:

1. Polynomial Functions: Always defined for all real numbers ($\mathbb{R}$). Example: $y = 2x - 3$
2. Radical Functions with Even Index: The expression inside the radical must be greater than or equal to zero ($f(x) \ge 0$). Example: For $y = \sqrt{x+2}$, the domain is $x+2 \ge 0$, meaning $x \ge -2$, or $[-2, +\infty)$.
3. Rational Functions: The denominator cannot be zero ($Q(x) \ne 0$). Example: For $y = \frac{1}{x+2}$, the domain is $x+2 \ne 0$, meaning $x \ne -2$, or $\mathbb{R} - \{-2\}$.

Questions & Discussion

Question 1: Does the function increase in the interval $(-3, -1)$? Based on the analysis of the provided graph in exercise 25, the function is indeed increasing in that specific interval.

Question 2: What about the interval $(-1, 1)$? In this interval, the function is decreasing, reaching a local minimum at $x = 1$.

Question 3: At what hours is the noise maximum and minimum according to the graph in exercise 28? The maximum noise level occurs at 16:00 (reaching $90\,dB$), and the minimum noise level occurs at 6:00 (reaching $50\,dB$). The graph shows noise levels staying constant between 0:00 and 8:00, then increasing until 16:00, decreasing until 18:00, and finally increasing again until 19:00.

Question 4: Is the value $-2.9$ included in the domain if the domain is $(-3, 2) \cup (3, 7)$? Yes, the value $-2.9$ is greater than $-3$ and less than $2$, so it falls within the first part of the union of intervals.