Definition
A function f : D → maps each x ∈ D to a unique f(x) ∈ . D is the domain.
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Lesson overview
This lesson brings together the PDF, study notes and chapter flow so you can revise General Functions without losing the thread.
Definition
A function f : D → maps each x ∈ D to a unique f(x) ∈ . D is the domain.
Image and preimage
f(x) is the image of x. If f(a) = b, then a is a preimage of b.
Parity
Even function: f(−x) = f(x) (symmetric about Oy). Odd: f(−x) = −f(x) (symmetric about O).
Increasing function
f is increasing on I if: ∀x₁, x₂ ∈ I, x₁ < x₂ ⟹ f(x₁) < f(x₂).
Maximum and minimum
f has a maximum M at x₀ if f(x₀) ≥ f(x) for all x in the domain.
Affine function f(x) = ax + b
Line with slope a and y-intercept b. Increasing if a > 0.
Square function f(x) = x²
Parabola, minimum at x=0. Decreasing on (−∞,0], increasing on [0,+∞).
Reciprocal function f(x) = 1/x
Defined on . Decreasing on (−∞,0) and on (0,+∞).
Square root f(x) = √x
Defined on [0,+∞). Increasing. (√x)² = x.
Theorem 1 - General Functions
Application of theorem 1.
Theorem 2 - General Functions
Application of theorem 2.
Theorem 3 - General Functions
Application of theorem 3.
Theorem 4 - General Functions
Application of theorem 4.
Theorem 5 - General Functions
Application of theorem 5.
Problem-solving method
Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.
Common mistakes
Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.