九至十二年级Analysis计划中
Functions and their graphs (precalculus)
Families of real functions (linear, quadratic, polynomial, rational, absolute value, piecewise), their graphs and transformations, and composition and inverses of functions of a real variable.
math.analysis.functions-and-graphs
本主题的课程尚未编写。下面的提纲是课程地图的一部分，尚未经过审校。
前置知识
FunctionsLinear relationships and simultaneous equationsQuadratic equations and functionsGraphs of change
23 项必需的前置知识（一直追溯到底）
你将学会
- Sketch graphs of standard function families and of transformations y = a f(b(x − h)) + k.
- Compose real functions, find natural domains, and find inverse functions on intervals where a function is injective.
- Identify zeros, asymptotes, symmetry and end behaviour of polynomial and rational functions.
定义
- The graph of f is the set {(x, f(x)) : x in the domain of f}.
- f is even if f(−x) = f(x), and odd if f(−x) = −f(x), for all x in a domain symmetric about 0.
- An inverse of f : A → B is a function g : B → A with g(f(x)) = x for all x ∈ A and f(g(y)) = y for all y ∈ B.
定理
- The graph of f⁻¹ is the reflection of the graph of f in the line y = x.
- A function has an inverse if and only if it is a bijection; a strictly monotone function on an interval is injective.
- The graph of y = f(x − h) is the graph of y = f(x) shifted h units to the right.
常见误解
- Writing f⁻¹(x) = 1/f(x).
- Thinking y = f(x − 2) shifts the graph to the left.
- Believing composition is commutative; in general f ∘ g ≠ g ∘ f.
通向
可选发现
- Composition Composition of functions is associative and has identities, the first axioms of a category.