The study of the signs of a function is one of the most important topics in mathematics and is frequently addressed in Enem tests. The function is one of the basic tools used in almost all branches of mathematics. It is a rule that relates each element of a set to a single element of another set. The study of functions and their signs is fundamental to understanding many mathematical concepts and solving a wide range of problems.
Before we go into detail about studying the signs of a function, it is crucial to understand what a function is. In simple terms, a function is a rule that relates each element of a set to a single element of another set. The set of all elements that can be input into the function is called the domain, while the set of all possible outcomes is called the range. Each input has exactly one output, which is the value of the function at that point.
In studying the signs of a function, we are interested in knowing where the function is positive (above the x-axis) and where it is negative (below the x-axis). This is especially important when we are solving inequalities involving functions. To determine the signs of a function, we need to find its zeros (the values of x for which the function equals zero) and check the sign of the function in each interval determined by these zeros.
To illustrate, consider the function f(x) = x^2 - 3x - 4. To find the zeros of this function, we solve the equation x^2 - 3x - 4 = 0. The solutions of this equation are x = - 1 and x = 4. Therefore, the zeros of the function are -1 and 4. Now, to determine the signs of the function, we choose a number from each of the intervals (-∞, -1), (-1, 4) and ( 4, ∞) and substituted into the function. For example, choosing x = -2, x = 0 and x = 5, we get f(-2) = 12, f(0) = -4 and f(5) = 6. Therefore, the function is positive on the interval (-∞ , -1), negative in the range (-1, 4) and positive in the range (4, ∞).
It is important to note that the study of the signs of a function is not limited to polynomial functions. We can apply the same process to rational functions, exponential functions, logarithmic functions, etc. Also, in some cases, we may need to consider not only where the function equals zero, but also where it is undefined.
In summary, studying the signs of a function is an essential skill in mathematics. It allows us to determine where a function is positive or negative, which is crucial for solving a variety of problems, especially those involving inequalities. To master this skill, it's important to practice with a variety of functions and problems. Fortunately, there are many resources available, including textbooks, math websites, and dedicated teachers or tutors, that can help make studying the signs of a function a less daunting task.
