Functions and their graphs (precalculus)

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 23

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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.

مفاهيم خاطئة شائعة

• 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.

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اكتشافات اختيارية

• Composition Composition of functions is associative and has identities, the first axioms of a category.

مسارات التعلّم

• Conventional K–12 progression
• Accelerated secondary progression
• Applied mathematics