Core functions, calculus, and algebra concepts from the IB Mathematics: Analysis and Approaches syllabus. Front: the term or formula. Back: definition or formula with a brief usage note.
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- Domain of a function
- The set of all possible input values (x-values) for which the function is defined.
- Range of a function
- The set of all possible output values (y-values) that the function can produce.
- Inverse function f^(-1)(x)
- A function that reverses the effect of f(x); if f(a) = b, then f^(-1)(b) = a. Exists only if f is one-to-one.
- Composite function (f o g)(x)
- The function f(g(x)), where the output of g becomes the input of f. Order matters: (f o g) is not the same as (g o f).
- Polynomial function
- A function of the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer and coefficients are real numbers.
- Rational function
- A function expressed as the ratio of two polynomials: f(x) = P(x)/Q(x), where Q(x) is not the zero polynomial.
- Vertical asymptote
- A vertical line x = a where the function approaches positive or negative infinity as x approaches a. Occurs when the denominator of a rational function equals zero.
- Horizontal asymptote
- A horizontal line y = b that the function approaches as x approaches positive or negative infinity. Found by comparing degrees of numerator and denominator in rational functions.
- Exponential function
- A function of the form f(x) = a*b^x, where a is non-zero, b is positive and b is not 1. Base b determines growth (b > 1) or decay (0 < b < 1).
- Logarithm definition
- If b^y = x, then log_b(x) = y. The logarithm is the exponent to which base b must be raised to get x.
- Change of base formula
- log_b(x) = log_a(x) / log_a(b). Allows conversion between logarithms of different bases; commonly used with natural log or base 10.
- Pythagorean identity
- sin^2(theta) + cos^2(theta) = 1. Fundamental trigonometric identity derived from the Pythagorean theorem on the unit circle.
- Sine rule
- a / sin(A) = b / sin(B) = c / sin(C), where a, b, c are side lengths opposite to angles A, B, C in a triangle. Used to find unknown sides or angles.
- Cosine rule
- c^2 = a^2 + b^2 - 2ab*cos(C), where c is the side opposite angle C. Generalizes the Pythagorean theorem for any triangle.
- Limit of a function
- The value that f(x) approaches as x approaches a value a. Written as lim(x->a) f(x) = L if f(x) gets arbitrarily close to L as x gets close to a.