Generally speaking, for two functions y=f(u) and u=g(x), if y can be expressed as a function of x through the variable u, then this function is called a composite function of functions y=f(u) and u=g(x), and is denoted as y=f(g(x)). When solving definite integral, we can use the following formula: ∫f(g(x))g'(x)dx=∫f(u)du.
Knowledge expansion
Function is a mathematical concept, which was first translated by China mathematician Li in the book Algebra. He translated this way because "any variable in this variable is a function of that variable", that is, a function means that one quantity changes with another quantity, or that one quantity contains another quantity.
The modern definition of function is: for a given number set A, assuming that the elements in it are X, there is a corresponding rule F, which is marked as f(x), so that each element X in A can be mapped to an element Y in another number set B through F, and the equivalent relationship between element X and its corresponding element Y can be expressed as y=f(x).
The concept of function can be expressed by the following formula:
Traditional definition: If there are two variables X and Y in a change process, and Y has a unique fixed value corresponding to each fixed value of X, then Y is said to be a function of X and X is an independent variable.
Modern definition: Let X be a nonempty set, Y a nonempty number set and F a corresponding rule. If there is only one element Y corresponding to any X in the corresponding rules F, Y and X, then F is called a function from set X to set Y. ..
Mapping definition: Let A and B be two non-empty sets. If there is a rule F, so that for each element X in A, there is a unique pixel Y in B according to the F correspondence rule, then F is called the mapping from A to B..
Simple definition: Let A and B be two nonempty sets. If an element X in A has a relationship F with an element Y in B, mathematically speaking, X has a relationship F with Y. ..
There are many kinds of functions, such as linear function, quadratic function, inverse proportional function, direct proportional function, exponential function, logarithmic function and so on. Each function has its own specific form and properties. For example, the expression of a linear function is y=kx+b, the expression of a quadratic function is y = ax 2+bx+c, the expression of an inverse proportional function is y=k/x, and so on.
In a word, function is a mathematical concept involving changes and relationships, and it is one of the most important basic concepts in mathematics. Through the study and research of functions, we can better understand and master the basic knowledge and application skills of mathematics.