A function is a fundamental concept in mathematics. It's a special type of relation between two sets where each input has exactly one output. Think of it as a machine that takes an input and produces a unique output. The key rule is that no input can produce multiple outputs, though multiple inputs can produce the same output.
Function notation provides a standardized way to express mathematical relationships. The notation f of x equals y tells us that function f takes input x and produces output y. We can evaluate functions by substituting specific values. For example, if f of x equals 2x plus 1, then f of 3 equals 7. The notation f colon A arrow B means function f maps set A to set B.
函數是數學中描述輸入與輸出關係的重要概念。它的核心特性是每個輸入值都對應唯一的輸出值。我們通常用f(x)來表示函數，其中x是輸入，f(x)是相應的輸出。例如，函數f(x)等於x的平方，當輸入x等於1時，輸出就是1的平方等於1。
函數的正式定義是從集合A到集合B的對應關係，其中A稱為定義域，B稱為值域。關鍵特性是定義域中每個元素都有唯一的輸出。這個映射圖展示了函數f(x)等於x的平方，將1映射到1，2映射到4，3映射到9。每個輸入都有唯一確定的輸出，這正是函數的本質。
函數可以用多種方式視覺表示。映射圖用箭頭顯示輸入如何連接到輸出。坐標圖在x-y平面上繪製點和曲線。數值表將輸入輸出對組織成行和列。垂直線測試很重要：如果任何垂直線與圖形相交超過一次，那它就不是函數，因為一個輸入會有多個輸出。
函數有許多不同類型。線性函數形式為mx加b，圖形是直線。二次函數包含x的平方項，圖形是拋物線。指數函數以某個數為底的x次方，表現出快速增長或衰減。對數函數是指數函數的反函數。三角函數如正弦和余弦具有周期性。每種函數類型都有其獨特的性質和應用。
函數在現實世界中有廣泛應用。在物理學中，函數描述位置、速度和力的關係。經濟學用函數分析供需和成本收益。工程學應用於信號處理和系統設計。計算機科學中函數是算法的基礎。日常生活中，溫度隨時間變化就是一個函數的例子，如圖所示的一天中溫度的周期性變化。
Let's explore common function types with examples. Linear functions like f of x equals 2x plus 1 create straight lines with constant rates of change. Quadratic functions such as f of x equals x squared form parabolic curves. Absolute value functions like f of x equals absolute value of x create V-shaped graphs. We can evaluate functions by substituting values: f of 2 equals 5, and f of negative 1 equals negative 1.
Domain and range are fundamental concepts for understanding functions. The domain is the set of all possible input values where the function is defined. The range is the set of all possible output values. For f of x equals x squared, the domain includes all real numbers, but the range is only y greater than or equal to zero since squares are never negative. For the square root function, both domain and range are restricted to non-negative values.