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Understand the concept of a function, its notation, domain, range, and different types of functions.

Input to Output: Functions

A function is a rule that assigns exactly one output to each input — written as f(x), where x is the input and f(x) is the output. Functions are the language of mathematics for describing how one quantity depends on another, from simple relationships like doubling a number to complex models in science, engineering, and economics.

Components of Functions

This section covers the foundational concepts of functions:

  • Function Notation: Using f(x) to name a function and evaluate it by substituting a value for x.
  • Domain & Range: The domain is the set of all valid inputs; the range is the set of all possible outputs.
  • Linear vs Nonlinear Functions: Linear functions produce straight-line graphs (constant rate of change); nonlinear functions produce curves.
  • Function Operations: Combining functions through addition, subtraction, multiplication, division, and composition (f(g(x))).

Examples of Functions

Function Notation Examples

  • If f(x) = 2x + 3, find f(4): Substitute 4 for x to get 2(4) + 3 = 11.
  • If g(x) = x² - 1, find g(-3): Substitute -3 to get (-3)² - 1 = 9 - 1 = 8.
  • If h(x) = 5x, then h(0) = 0, h(1) = 5, and h(2) = 10 — each input produces exactly one output.

Domain & Range Examples

  • For f(x) = x + 4, the domain is all real numbers and the range is all real numbers.
  • For f(x) = √x, the domain is x ≥ 0 (no square root of negatives) and the range is f(x) ≥ 0.
  • For f(x) = 1/x, the domain is all real numbers except x = 0 (division by zero is undefined).

Linear vs Nonlinear Examples

  • f(x) = 3x - 2 is linear: every increase of 1 in x increases f(x) by 3. The graph is a straight line with slope 3.
  • f(x) = x² is nonlinear: the rate of change increases as x grows. The graph is a parabola opening upward.
  • A car traveling at a constant 60 mph follows a linear function d(t) = 60t, while a ball falling under gravity follows a nonlinear function.

Function Operations Examples

  • If f(x) = x + 1 and g(x) = 2x, then (f + g)(x) = x + 1 + 2x = 3x + 1.
  • If f(x) = x² and g(x) = x + 3, then f(g(x)) = f(x + 3) = (x + 3)² = x² + 6x + 9.
  • If f(x) = 4x and g(x) = x/2, then f(g(x)) = 4(x/2) = 2x, and g(f(x)) = 4x/2 = 2x — these functions are inverses since both compositions return x when simplified further with the right pair.