Understanding Functions

• Functions relate two numerical variables commonly labeled x and y.
• 'x' refers to the independent variable.
• 'y' is the dependent variable.
• The function associates each value of x with a single value of y, expressed as y = f(x). A graphical representation is used to visualize the behavior of a function.

Key Concepts in Functions

• A function relates two variables.
• Functions can describe natural phenomena.
• Functional relationships can be described using formulas.
• Functions can be represented graphically.

Historical Contributions

• Nicolas de Oresme (14th century) was the first to describe laws of nature as relationships of dependency between two variables.
• Galileo (16th century) used experimentation to establish numerical relationships.
• Descartes (17th century) enabled functions to be graphically represented with his algebraization of geometry.
• Leibniz (17th century) first used the word "function" to describe these types of relations in 1673.
• Euler (18th century) refined the concept, providing precision and generality, and introduced the notation F(x).
• Dirichlet (19th century) broadened the function concept, allowing relations between two variables to be functions even without an analytical expression.
• The domain of a function 'f' (Dom f) is the set of x values for which the function exists.
• The range of 'f' is the set of values that the function takes.

Function Representation and Analysis

• Functions can be presented through graphs, tables of values, formulas, or verbal descriptions.
• The graphical representation helps appreciate the overall behavior of a function.
• Numerical data, combined with an enunciation, can define the function precisely.

Analytical Expression or Formula

• Analytical expressions offer the most precise way to define a function, but they require detailed analysis for visualization.

Cuts with the axes/Sign of Function

• The points where a function intersects the x-axis usually indicate a change in the sign of its values.
• Knowing these points is important.
• When a function is given by an analytical expression y = f(x), finding the intersection points with the axes is straightforward.
• To find where the function intersects the x-axis, set f(x) = 0.
• A function can only intersect the y-axis once, found by calculating f(0).

Function Sign

• The sign of a function can be either positive or negative.
• Solve two inequalities:
  • f(x) > 0, for intervals where f(x) is positive.
  • f(x) < 0, for intervals where f(x) is negative.
• A function can be continuous in an interval [a, b] if it does not present any discontinuities in it.
• Users favor fees that are governed by graphs.

Function Variation

• The average height of people is a rising function.
• Sleep is a decreasing function.
• Functions can be analyzed from left to right, observing how the values of 'y' evolve as the values of 'x' increase.
• A function 'f' is increasing if when x₂ > x₁, then f(x₂) > f(x₁).
• A function 'f' is decreasing if when x₂ > x₁, then f(x₂) < f(x₁).
• A function can be increasing in intervals and decreasing in others. The function is defined in the interval [-7, 11].
• Increasing in the intervals (-7, -3) and (1, 11).
• Decreasing in the interval (-3, 1).

Maxima and Minima

• A function has a relative maximum at a point.
• Its value at this point is greater than the surrounding points.
• If 'f' has a relative minimum at a point, it decreases before the point and increases after it.
• It can take on larger values than a relative height.

Variability

• To measure the rate of change of a function in a data series/equation, use the average rate of change.
• If f is growing it has a positive rate, and if the function decreases, the slope is negative.

Tendency and Periodicity

• A paratrooper stabilizes.

Function limit

• These trends are known as function limits.

Periodicity

• It is represented by the variation in height.
• In this function, what happens in 30 segs repeats.
• The length of a periodic function is called a period.