Relation between equilibrium constant and energy

Real valued function

A function f : A → B is said to be a real-valued function if A ⊆ R, B ⊆ R.

Usually a function will be given by its relation y = f(x) between the elements x ∈ A and y ∈ B where A is the domain and B is the codomain of f.

If y = f(x) is a function, then x is called independent variable and y is called dependent variable.

For example y = ax + b is a function.

If the domain of a function is not specified (given), then its domain will be taken as maximum possible subset A of R such that y = f(x) is defined for all x ∈ A.

Types of Functions:

(i) If f is a real valued function, then for each real number x there exists a unique real number y such that y = f(x).
Then x is called independent variable and y is called dependent variable.
The function f whose relation between the variables is of the form y = f(x), is called explicit function

Ex: y = x2 + 1; y = log x

(ii) If the relation between the independent variable x and the dependent variable y of a function is of the form f(x, y) = 0, then such a function is called implicit function.

Ex: x2 + xy + y2 + 1 = 0; x3 + 7x2y + 2xy2 + y3 – 5x = 0

(iii) Let f : A → R be a function. Then
  • f is said to be monotonically increasing on A if
    x1, x2 ∈ A, x1 < x2 ⇒ f(x1) ≤ f(x2)
  • f is said to be strictly increasing on A if
    x1, x2 ∈ A, x1 < x2 ⇒ f(x1) < f(x2)
  • f is said to be monotonically decreasing on A if
    x1, x2 ∈ A, x1 < x2 ⇒ f(x1) ≥ f(x2)
  • f is said to be strictly decreasing on A if
    x1, x2 ∈ A, x1 < x2 ⇒ f(x1) > f(x2)
  • f is said to be monotonic on A if
    f is either monotonically increasing or monotonically decreasing on A.
(iv) A function f : A → R is said to be "bounded on A" if there exists real numbers k1, k2 such that k1 ≤ f(x) ≤ k2, ∀ x ∈ A. i.e, | f(x) | ≤ k ∀x ∈ A