disjoint: two sets have no elements in common at all
Surds are numbers constructed from applying a rational number power on a rational number. \[ Surds = \{p^q | p,q \in Q, p\geqslant0.\}\] The surds include all rational numbers but not vice versa
The set of numbers which are not surd.
Surds + transcendental numbers = real numbers
\[ log_{m}a = \frac{log_{n}a}{log_{n}m} \quad for\quad n\ne 0\]
Injective: one-to-one \[ for \quad all \quad x_{1},x_{2} \in X, \quad then \quad f(x_{1}) = f(x_{2}) \Longleftrightarrow x_{1} = x_{2} \]
surjective: onto \[ for \quad y \in Y, there \quad exist \quad x \in X \quad such \quad that \quad f(x) = y \\ i.e. \quad Codomain = Range \]
bijective: if a function is both injective and surjective
Monotone increasing/decreasing: \[ for \quad all \quad x_{1},x_{2} \in X, \quad such \quad that \quad x_{1} < x_{2} \quad then \quad f(x_{1}) \le (\ge) f(x_{2})\]
Strictly increasing/decreasing: \[ for \quad all \quad x_{1},x_{2} \in X, \quad such \quad that \quad x_{1} < x_{2} \quad then \quad f(x_{1}) < (>) f(x_{2})\]
\[ P(x) = a_{n}x^n + a_{n-1}x^{n-1} + ... + a_{2}x^2 + a_{1}x + a_{0} \] * Power function
\[ f(x) = x^a, \quad where \quad a \quad is \quad a \quad constant \]
\[ f(x) = \frac{P(x)}{Q(x)} \]
Algebraic functions: a functionn is called an algebraic function if it can be constructed using algebraic operations (such as addition, division, and taking roots) starting with polynomials
Exponential functions:
\[ f(x) = b^x, where \quad b \quad is \quad a \quad positive \quad constant\]
Let f(x) be a real polynomial. The remainder of f(x) when divided by the expression (x-a) is given by f(a) \[ \frac{f(x)}{x-a} = g(x) + f(a) \] real polynomial: the coefficients are real numbers
Let f(x) be a real polynomial. The expression (x-a), where \(a \in R\) is a factor of f(x) if and only if f(a)=0