Calculus

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A function is an expression, rule, or law that defines a relationship between one variable (independent variable) and another variable (dependent variable). In a function, each input value must have o…

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Domain: The set of all possible input values (x-values) for which the function is defined. Range: The set of all possible output values (y-values) that the function can produce. For f(x) = √(x+4): Do…

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A polynomial function is defined as f(x) = a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² + ... + aₙ₋₁x + aₙ where a₀, a₁, a₂, ..., aₙ are real numbers, n is a non-negative integer, and a₀ ≠ 0. Example: f(x) = 3x³ - 2x² + 5…

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An exponential function has the form f(x) = aˣ where a > 0 and a ≠ 1. Graph behavior: - If a > 1: Function is strictly increasing - If 0 < a < 1: Function is strictly decreasing - If a = 1: Graph is …

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The Greatest Integer Function is defined as f(x) = ⌊x⌋, where ⌊x⌋ is the greatest integer less than or equal to x. Examples: ⌊2.7⌋ = 2 ⌊-1.3⌋ = -2 ⌊5⌋ = 5 The graph consists of horizontal line segme…

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The Modulus Function is defined as: f(x) = |x| = {x if x ≥ 0; -x if x < 0} It gives the absolute value of x. Graph characteristics: - V-shaped graph with vertex at origin (0,0) - Right side: y = x (…

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The Signum Function is defined as: sgn(x) = {-1 for x < 0; 0 for x = 0; +1 for x > 0} It indicates the sign of a real number: - Returns -1 for negative numbers - Returns 0 for zero - Returns +1 for p…

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The limit of a function f(x) as x approaches a value 'a' is the value that f(x) gets arbitrarily close to as x gets closer and closer to 'a'. Symbolic representation: lim[x→a] f(x) = L This means: A…