Functions

This chapter is the starting point of higher mathematics. We will systematically study the three core concepts: functions, limits, and continuity. They are not only the foundation for learning calculus but also key tools for describing and solving practical problems using mathematical language.

Functions

A function is a fundamental mathematical model that describes the dependency relationship between variables.

Concept and Representation of Functions

Definition: Given two variables x and y, if for every value of x in a certain range D, there is a unique value of y corresponding to it, then y is called a function of x, denoted as $y = f(x)$. Here, x is called the independent variable, y is the dependent variable, and D is the domain of the function.

Representation Methods:

1. Analytical (Formula) Method: Use a mathematical expression to represent the function relationship, such as $y = x^2 + 1$. This is the most common method.
2. Tabular Method: List the independent variable and the corresponding function values in a table.
3. Graphical Method: Use a curve on the coordinate plane to represent the function relationship.

Properties of Functions

Boundedness

• Definition: If there exists a constant M such that for any $x \in D$, $|f(x)| \leq M$, then $f(x)$ is said to be bounded on D.

Monotonicity

• Definition: Let $f(x)$ be defined on interval I.
  • If for any $x_1, x_2 \in I$, when $x_1 < x_2$, $f(x_1) < f(x_2)$ always holds, then $f(x)$ is said to be increasing on I.
  • If for any $x_1, x_2 \in I$, when $x_1 < x_2$, $f(x_1) > f(x_2)$ always holds, then $f(x)$ is said to be decreasing on I.

Periodicity

• Definition: If there exists a nonzero constant T such that for any x in the domain, $f(x + T) = f(x)$ always holds, then $f(x)$ is called a periodic function, and T is called its period. Usually, we refer to the smallest positive period.

Parity

• Prerequisite: The domain of the function is symmetric about the origin.
  • Even Function: If $f(-x) = f(x)$, then $f(x)$ is an even function, and its graph is symmetric about the y-axis.
  • Odd Function: If $f(-x) = -f(x)$, then $f(x)$ is an odd function, and its graph is symmetric about the origin.

Special Types of Functions

1. Composite Function: If $y = f(u)$ and $u = g(x)$, then $y = f[g(x)]$ is called the composite function of $f$ and $g$.
2. Inverse Function: If $y = f(x)$ is monotonic, then it has an inverse function, denoted as $x = f^{-1}(y)$ or $y = f^{-1}(x)$. The graphs of the original and inverse functions are symmetric about the line $y = x$.
3. Piecewise Function: A function represented by different expressions in different parts of its domain.
4. Implicit Function: A function relationship determined by the equation $F(x, y) = 0$.

Elementary Functions

• Basic Elementary Functions: Include power functions, exponential functions, logarithmic functions, trigonometric functions, and inverse trigonometric functions.
• Elementary Functions: Functions obtained by a finite number of algebraic operations and compositions of basic elementary functions.

Exercises

1. Find the domain of the function $f(x) = \frac{1}{\sqrt{4 - x^2}} + \ln(x)$.
2. Determine the parity (odd/even) of the function $f(x) = \ln(\frac{1 - x}{1 + x})$.
3. Find the inverse function of $y = e^{2x} + 1$.