什么是函数?
代数是一门数学学科,处理符号和可能对它们执行的操作。这些符号被称为变量,因为它们没有任何设定值。我们经常遇到在现实生活问题中发生变化的特定价值观。然而,表达这些移动值的必要性是不变的。这些值在代数中通常用 x、y、z、p 或 q 等符号表示,这些符号称为变量。此外,这些符号经过各种算术运算,包括加法、减法、乘法和除法,目的是确定值。
Algebra allows you to articulate issues or circumstances as mathematical expressions. To construct a meaningful mathematical statement, it uses variables like x, y, and z, as well as mathematical operations like addition, subtraction, multiplication, and division. Algebra is used in all fields of mathematics, including trigonometry, calculus, and coordinate geometry. 2x + 4 = 8 is a simple example of an algebraic expression.
几个代数表达式的使用降低了代数的复杂性。根据表达式的使用方式和复杂程度,代数可以分为几个分支。下面列出了这些分支:
- 预代数
- 代数 I(初级)
- 抽象形式的代数
- 代数(通用)
什么是函数?
A function is a method or a relationship that connects each member ‘a’ of a non-empty set A to at least one element ‘b’ of another non-empty set B. In arithmetic, a function is a relation f from one set A (the domain of the function) to another set B (the co-domain of the function).
In mathematics, functions are the most fundamental aspect of calculus. Functions are certain forms of relationships. In arithmetic, a function is represented as a rule that produces a unique output for each input x. In mathematics, the term “mapping” or “transformation” is used to describe a function. Letters like f, g, and h are commonly used to represent these functions. The domain is defined as the collection of all possible input values for the function when it is defined. The range refers to all of the values that the function’s output produces. The collection of values that could be outputs of a function is known as the co-domain.
- If every element of set A has exactly one image in set B, the relation is said to be a function.
- A function is a relation from a non-empty set B with domain A and no two separate ordered pairs in f having the same initial element.
- If f(a) = b is a function from A B and (a,b) f, then f(a) = b is a function from A B and (a,b) f, where ‘b’ is the image of ‘a’ under ‘f’ and ‘a’ is the preimage of ‘b’ under ‘f’.
- If a function f: A B exists, set A is referred to as the function’s domain, and set B is referred to as the function’s co-domain.
函数类型
在算术中,函数非常重要,因为它们允许我们探索多种函数。我们有四个基于从集合 A 到集合 B 的元素映射的函数。
- 内射函数或一对一函数:当两个集合之间的每个域的范围存在映射时。
- f: A → B 称为一对一或单射的,如果 A 在 f 下的不同元素的图像是不同的,即对于 A 中的每个 a,b,f(a) = f(b), ⇒ a = b。否则,它是多对一的。
- 满射函数或 Onto函数:当有多个元素从域映射到范围时。
- f:如果 f 既是一对一又是上,则称 A → B 是一对一和上或双射的。
- f: A → B 被称为在,如果 B 的每个元素都是 A 在 f 下的某个元素的图像,即对于每个 b ϵ B,在 A 中存在一个元素 a 使得 f(a) = b .当且仅当函数的范围 = B 时,函数是 on。
- 多项式函数:由多项式组成的函数。
- 反函数:函数反转另一个函数的函数。
图上的函数
知道 x 的值允许在图形上表示函数f(x)。因为 y = f(x),我们可以从 x 的值开始找到 y 的相关值。因此,我们可以使用 x 和 y 值在坐标平面中绘制图形。考虑以下场景:
假设 y = x + 3
当 x = 0 时,y = 3;
相似地,
- x = -2,y = -2 + 3 = 1
- x = -1,y = -1 + 3 = 2
- x = 1,y = 1 + 3 = 4
- x = 2,y = 2 + 3 = 5;
- x = 3, y = 2 + 3 = 6
因此,我们可以使用这些值绘制函数x + 3 的图形。
函数的应用
当我们说变量 y 是变量 x 的函数时,我们表示 y 依赖于 x,并且 y 的值由 x 的值决定。这种依赖关系可以表示如下:f = y (x)。
- 圆的半径 A = r2 可用于计算圆的面积。半径 r 影响面积 A。我们在函数的数学语言中声明 A 是 r 的函数。
- 球体的体积 V 是其半径的函数。 V = 4/3 r3 表示 V 对 r 的依赖性。
- 力是质量固定为 m 的物体的加速度的函数。
类似问题
问题1:函数的两个例子是什么?
回答:
Some examples of functions would be linear functions: f(x) = ax+b, or polynomial functions:
f(x) =anxn + … + a1x + a0
There are many others such as quadratic, cubic, rational, rational, trigonometric, etc.
问题2:求函数g(t)= 6t 2 +5g(t)=6t 2 +5 at
(一) t = 0
(ii) t = 2
回答:
The given function is g(t)= 6t2+5g(t)=6t2+5
(i) At t = 0,
g(0)= 6(0)2+5g(0) = 6(0)2+5
= 5
(ii) At t = 2, g(2) = 6(2)2+5g(2)=6(2)2+5
= 29
问题3:代数函数有哪些不同类型?
回答:
Linear functions, quadratic functions, cubic functions, polynomial functions, radical functions, and rational functions are examples of algebraic functions. f(x) =2x+3 (linear), f(x)=(2x+3)/(x2) (rational), and f(x)=x(1/2) are some instances (rational).
问题4:什么是线性函数?
回答:
A linear function is an algebraic equation in which each term is a constant or a product of constants plus a single power 1 independent variable.