Grades 9–12AnalysisPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
FunctionsLinear relationships and simultaneous equationsQuadratic equations and functionsGraphs of change
Required prerequisites, all the way down: 23
You will learn to
- 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.
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Leads to
Optional discoveries
- Composition Composition of functions is associative and has identities, the first axioms of a category.