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, '4x ** 3 - 3x ** 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)')

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 'ax ** 2 + bx + c = 0')
• polynomial equations: higher-degree equations (eg 'x ** 3 - 4x ** 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 'lox(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 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: ''
• exponential functions: ''
• logarithmic functions: ''
• trigonometric functions: ''

• piecewise functions

  • defined by different expressions over different intervals
  • ''
• implicit functions: ''

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: ''
• rational inequalities: ''
• exponential inequalities: ''
• logarithmic inequalities: ''
• trigonometric inequalities: ''

the number of arguments or inputs a function takes.

• types:

  • unary: a function with one input (eg '')

  • binary: a function with two inputs (eg '')

  • n-ary: a function with nn inputs (eg '')

the set of all possible inputs for the function.

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

the set of all possible outputs of the function.

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

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' even if the range is only non-negative reals.

a function ff is injective if different inputs map to different outputs.

example: ''

a function ff is surjective if every element in the codomain is mapped by some element in the domain.

example: ''

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

example: ''

a function ff is continuous if small changes in the input result in small changes in the output.

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

a function ff is differentiable if it has a derivative at each point in its domain.

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

a function ff is periodic if there exists a positive number tt such that f(x+t)=f(x)f(x+t)=f(x) for all xx.

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

• 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'

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

types:

• monotonically increasing: ''
• monotonically decreasing: ''

a function ff is bounded if there exists a number mm such that ∣f(x)∣≤m∣f(x)∣≤m for all xx in the domain.

types:

• bounded above: ''
• bounded below: ''

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'

• asciimath

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