X is the domain of a function and Y is the range co-domain.
f : X \to Y
A function is injective if for any two arguments x_1, x_2 \in X we have: f(x_1) = f(x_2) \implies x_1 = x_2. In other words: a given y can be obtained only through one x.
(\forall y \in Y)(\exists x \in X)(f(x) = y). In other words: All y \in Y have to be obtainable.
The inverse of a function is denoted by f^{-1}: Y \to X. The function has to be surjective and injective to be invertible: f^{-1}(b) = a if f(a) = b. Its also true that for (f^{-1} \circ f)(x) = (f \circ f^{-1})(x) = x1
For some set A, the image of A by f is f(A) = \{f(x): x \in A\}. We can also define an inverse of an image even when the function itself isn’t invertible f^{-1}(A).
\log_a(b) = c \equiv a^c = b where a, b > 0\ \cap a \neq 1\ \cap b \neq 1
\log_a(x^c) = c\log_a(x)
\log_a(b) = \frac{1}{\log_b(a)}
\log_{a^n}(b) = \frac{1}{n}\log_a(b)
\log_a(b) = \frac{\log_c(b)}{\log_c(a)}
The horizontal asymptote has a form of y = b where:
\lim_{x \to \infty} f(x) = b or \lim_{x \to -\infty} f(x) = b
The vertical line x = a is an asymptote if:
\lim_{x \to a^-} f(x) = \pm \infty or \lim_{x \to a^+} f(x) = \pm \infty
A line y = ax + b is an oblique asymptote at \infty where:
\begin{matrix} a = \lim_{x \to \infty} \frac{f(x)}{x} \\ b = \lim_{x \to \infty} (f(x) - ax) \\ \end{matrix}
A function is said to be continuous if at all points it is true that: f(x_0) = \lim_{x \to x_0^-}f(x) = \lim_{x \to x_0^+}f(x)
A function is said to be differentiable at some point x_0 if the f'(x_0) exists. It also means that the function is continuous at point x_0 but not the other way around.
To find an extremum we can take a derivative and compare to zero (Make sure the derivative at x_0 exists): f'(x_0) = 0. Then using the found arguments f(x_0) is an extremum.
A local maximum or minimum is a maximal or minimal value of a function in a given range
A global maximum or minimum is a maximal or minimal value of a function in Y
Composition operator: (g \circ f)(a) \equiv g(f(a))↩︎