Algebra → Linear Algebra · Lesson 2

Functions: From Algebra to Analysis

Functions are the central objects of modern math. Every formula, every graph, every algorithm computes a function. Understanding domain, range, composition, and inverses unlocks all higher math.

Key Concepts

Function Fundamentals

f(x) = rule applied to x. Domain = valid inputs. Range = possible outputs. Each input gives exactly one output. Test with vertical line test on a graph.

Key Function Families

Linear: f(x) = mx + b (straight line). Quadratic: ax² + bx + c (parabola). Exponential: aᵇˣ (growth/decay). Logarithmic: log_b(x) (inverse of exponential). Trig: sin, cos, tan (periodic).

Composition & Inverse

Composite (f∘g)(x) = f(g(x)) — apply g then f. Inverse f⁻¹ undoes f: f(f⁻¹(x)) = x. Find inverse by swapping x and y, then solving for y.

Even, Odd & Transformations

Even: f(−x) = f(x), symmetric about y-axis. Odd: f(−x) = −f(x), symmetric about origin. Transformations: f(x)+k (up k), f(x−h) (right h), af(x) (stretch a).

Live Python Practice

Interactive Lab

f(x): Show inverse f⁻¹(x)

Blue = f(x), Red dashed = f⁻¹(x). The inverse is the reflection over the y=x line.

Check Your Understanding

The domain of f(x) = ln(x) is:

If f(x) = x + 2, then f⁻¹(x) =

f(g(x)) means:

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