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Understanding Functions

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Function Definition

Let and be two non-empty subsets of the set of real variables . If there exists a definite rule which associates each element of of to a unique element of , then this rule is called a real valued function on the set of real variables and is denoted by the symbol . The element in corresponding to the element in obtained by the rule is denoted by . Thus, if the rule determines when , then we have, . represents a mathematical relation which determines uniquely , when .

The set of all real values over which varies is called the domain of definition of the function. The domain of definition of is .

Again, the set of all real values of or , for all , is called the range of the function. Thus, range of is .

If is a real valued function of , then the variable to which arbitrarily different values of the domain of definition are assigned is called the independent variable, while the variable which assumes values of the range of the function is called the dependent variable.

Example:

Suppose and are two real variables and they are connected by the mathematical equation, . then, will represent a real function as the relation assigns a unique value of , for every real value of . For example, when . Again, when . Clearly, the domain of definition of the function is and its range is .

Single-valued and Multiple-valued Functions:

If the law or rule of association between two real variables and is such that each value of x corresponds to a unique value of , then is called a single-valued function of x. However, if the law of association between them determines more than one value of for each given value of within its domain, then is called a multiple-valued function of .

For example, the function defines y as a multiple-valued(or double-valued) function of since every  positive real value of corresponds to two real values of .

A multiple-valued function is usually expressed as two or more single-valued functions by imposing conditions on the dependent variable. Thus, the multiple-valued function can be broken into two single-valued functions, viz., and .

Classification of Functions

1. Algebraic Functions

An algebraic expression in a variable containing a finite number of terms is called an algebraic function of that variable. In general, there are three types of algebraic functions:

a. Polynomial Function:

If is a positive integer and are real constants, then the expression

is called a polynomial function in of degree and is denoted by . Thus,

, where is real.

For example, latex x $ of degree 4, 3 and 1 respectively.

b. Rational Function:

The ration of two polynomial functions is called a rational function and is denoted by . If and are two polynomial functions, then,

[where is real and ]

represents a rational function.

Each of the functions represents a rational function.

c. Irrational Function:

An algebraic function which is not rational is called an irrational function.

Each of the functions is an irrational function.

2. Non-algebraic or Transcendental Functions

Functions which are not algebraic are called non-algebraic or transcendental functions. A few important non-algebraic functions are as follows:

a. Exponential Functions:

If is a real variable then the functions and are called exponential functions.

b. Logarithmic Functions:

If is a real variable, then each of the functions is called a logarithmic function.

c. Trigonometrical Functions:

The six functions where the angle is measured in radian, are called trigonometrical functions.

d. Inverse Circular Functions:

Th six functions , , , $latex \csc^{-1}{x} (x \geq 1 \: or \: x \leq -1) $ are called inverse circular functions.

3. Explicit and Implicit Functions

If the dependent variable can be expressed directly in terms of the independent variable , then the function is said to be explicit. For example, are explicit functions. In other words, is said to be an explicit function of if it can be expressed as .

If the dependent variable cannot be expressed directly in terms of the independent variable , then the function is said to be implicit. For example, are implicit functions. In other words,  is said to be an implicit function of if it is given in the form .

Sometimes, an implicit function can be reduced to an explicit function. For example, the implicit function may be written as,

which is the explicit form of the given function.However, the implicit function is not reducible to explicit form.

Parametric Form of Function

In parametric form of a function both the independent and the dependent variables are expressed in terms of a third variable. If the independent variable and the dependent variable of the function are expressed as functions of a third variable , i.e., if , then these two relations represent the parametric form of and  is called the parameter. The functional relation can be obtained by elimination of from these two relations.

Composite Functions or Functions of Function

Let us consider the following functions:

i)

ii)

In (i), is the exponential function of and which is a rational function of .

Similarly, in (ii) if we assume , then .

Clearly, is the logarithmic function of and is an irrational function of .

Hence, both and may be considered functions of another function.

Such functions like and are called composite functions.

Exercise

  1. Does the equation represent as a function of ? If so, find the domain of definition and range of the function.
  2. If , prove that:
  3. If , prove that,
  4. Find the domain of:
  5. Find the range of:
  6. If, , prove that,
  7. If , show that,
  8. If prove that,
  9. Find the value of for which the following functions are undefined:
  10. If express as an explicit function of .
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