# overview of mathematical structures
# expressions
expressions represent values and consist of variables, constants, and operators. they do not include an equals sign or assert equality.

* algebraic expressions:
  * monomial: a single term (eg '5 x')
  * binomial: two terms (eg 'x ** 2 + 2 x')
  * polynomial: multiple terms (eg '4 x ** 3 - 3 x ** 2 + 2 x - 1')
* rational expressions: ratios of polynomials (eg '(x ** 2 - 1) / (x + 1)')
* exponential expressions: expressions involving exponents (eg '2 ** x')
* logarithmic expressions: expressions involving logarithms (eg 'log(x)')
* trigonometric expressions: expressions involving trigonometric functions (eg 'sin(x) + cos(x)')
* radical expressions: expressions involving roots (eg 'sqrt(x + 1)')

# equations
equations state that two expressions are equal and include an equals sign. they can be solved to find the values of the variables.

* linear equations: first-degree equations (eg '2 x + 3 = 7')
* quadratic equations: second-degree equations (eg 'a x ** 2 + b x + c = 0')
* polynomial equations: higher-degree equations (eg 'x ** 3 - 4 x ** 2 + x + 6 = 0')
* rational equations: equations involving rational expressions (eg '1 / (x + 1) = 2')
* exponential equations: equations involving exponential expressions (eg '2 ** x = 8')
* logarithmic equations: equations involving logarithms (eg 'log(x) = 3')
* trigonometric equations: equations involving trigonometric functions (eg 'sin(x) = 0.5')
* differential equations: equations involving derivatives (eg '(d ** 2 * y) / (d x ** 2) + 3 * ((d y) / (d x)) + 2 y = 0')

# functions
functions describe a relationship between inputs and outputs. each input corresponds to exactly one output.

* linear functions: 'f(x) = m x + b'
* quadratic functions: 'f(x) = a x ** 2 + b x + c'
* polynomial functions: 'f(x) = x ** 3 - 4 x ** 2 + x + 6'
* rational functions: 'f(x) = (x ** 2 - 1) / (x + 1)'
* exponential functions: 'f(x) = 2 ** x'
* logarithmic functions: 'f(x) = log(x)'
* trigonometric functions: 'f(x) = sin(x) + cos(x)'
* piecewise functions:
  * defined by different expressions over different intervals
  * example: 'f(x) = x' if 'x >= 0', 'f(x) = -x' if 'x < 0'
* implicit functions: defined by relations not solved for the dependent variable
  * example: 'x ** 2 + y ** 2 = 1'

# inequalities
inequalities express a relationship where two expressions are not necessarily equal, but one is greater or less than the other.

* linear inequalities: '2 x + 3 > 7'
* polynomial inequalities: 'x ** 2 - 4 < 0'
* rational inequalities: '(x - 1) / (x + 2) >= 0'
* exponential inequalities: '2 ** x < 5'
* logarithmic inequalities: 'log(x) > 1'
* trigonometric inequalities: 'sin(x) <= 0.5'

# characteristics of functions
# arity
the number of arguments or inputs a function takes.

* types:
  * unary: a function with one input (eg 'f(x) = x ** 2')
  * binary: a function with two inputs (eg 'f(x, y) = x + y')
  * n-ary: a function with n inputs (eg 'f(x_1, ..., x_n) = x_1 + ... + x_n')

# domain
the set of all possible inputs for the function.

example: for 'f(x) = sqrt(x)', the domain is 'x >= 0'.

# range
the set of all possible outputs of the function.

example: for 'f(x) = x ** 2', the range is 'y >= 0'.

# codomain
the set of values that could potentially be outputs, not necessarily all of which are achieved.

example: for 'f: real_numbers -> real_numbers', the codomain is 'real_numbers'.

# injectivity (one-to-one)
a function 'f' is injective if different inputs map to different outputs.

example: 'f(x) = 2 x + 1' is injective.

# surjectivity (onto)
a function 'f' is surjective if every element in the codomain is mapped by some element in the domain.

example: 'f(x) = x ** 3' from 'real_numbers' to 'real_numbers' is surjective.

# bijectivity
a function is bijective if it is both injective and surjective.

example: 'f(x) = x + 5' from 'real_numbers' to 'real_numbers' is bijective.

# continuity
a function 'f' is continuous if small changes in the input result in small changes in the output.

example: 'f(x) = x ** 2' is continuous everywhere.

# differentiability
a function 'f' is differentiable if it has a derivative at each point in its domain.

example: 'f(x) = x ** 3' is differentiable everywhere.

# periodicity
a function 'f' is periodic if there exists a positive number 't' such that 'f(x + t) = f(x)' for all 'x'.

example: 'f(x) = sin(x)' has period '2 pi'.

# symmetry
* even function: 'f(-x) = f(x)' for all 'x' in the domain
  * example: 'f(x) = x ** 2'
* odd function: 'f(-x) = -f(x)' for all 'x' in the domain
  * example: 'f(x) = x ** 3'

# monotonicity
a function is monotonic if it is either entirely non-increasing or non-decreasing.

* monotonically increasing: 'f(x) = x'
* monotonically decreasing: 'f(x) = -x'

# boundedness
a function 'f' is bounded if there exists a number 'm' such that '|f(x)| <= m' for all 'x' in the domain.

* bounded above: 'f(x) = 1 / (1 + x ** 2)' is bounded above by 1
* bounded below: 'f(x) = x ** 2' is bounded below by 0

# inverse function
a function 'f' has an inverse 'f ** -1' if 'f((f ** -1)(y)) = y' and '(f ** -1)(f(x)) = x'

example: the inverse of 'f(x) = 2 x' is '(f ** -1)(x) = x / 2'