Domain and range

We have seen that a function is a special relation. In the same sense, real function is a special function. The special about real function is that its domain and range are subsets of real numbers “R”. In mathematics, we deal with functions all the time – but with a difference. We drop the formal notation, which involves its name, specifications of domain and co-domain, direction of relation etc. Rather, we work with the rule alone. For example,

$$f\left(x\right)=x^2+2x+3$$

This simplification is based on the fact that domain, co-domain and range are subsets of real numbers. In case, these sets have some specific intervals other than “R” itself, then we mention the same with a semicolon (;) or a comma(,) or with a combination of them :

$$f\left(x\right)=\sqrt{{\left(x+1\right)}^2-1};x<-2,x\ge 0$$

Note that the interval “ $x<-2,x\ge 0$ ” specifies a subset of real number and defines the domain of function. In general, co-domain of real function is “R”. In some cases, we specify domain, which involves exclusion of certain value(s), like :

$$f\left(x\right)=\frac{1}{1-x},x\ne 1$$

This means that domain of the function is $R-\left\{1\right\}$ . Further, we use a variety of ways to denote a subset of real numbers for domain and range. Some of the examples are :

• $x>1:$ denotes subset of real number greater than “1”.
• $R-\{0,1\}$ denotes subset of real number that excludes integers “0” and “1”.
• $1<x<2:$ denotes subset of real number between “1” and “2” excluding end points.
• $\left(\mathrm{1,2}\right]:$ denotes subset of real number between “1” and “2” excluding end point “1”, but including end point “2”.

Further, we may emphasize the meaning of following inequalities of real numbers as the same will be used frequently for denoting important segment of real number line :

• Positive number means x>0 (excludes “0”).
• Negative number means x<0 (excludes “0”).
• Non - negative number means x ≥ 0 (includes “0”).
• Non – positive number means x ≤ 0 (includes “0”).

Domain of real function

Domain of real function is a subset of “R” such that rule “f(x)” is real. This concept is simple. We need to critically examine the given function and evaluate interval of “x” for which “f(x)” is real.

In this module, we shall restrict ourselves to algebraic functions. We determine domain of algebraic function for being real in the light of following facts :

• If the function has rational form p(x)/q(x), then denominator q(x) ≠ 0.
• The term √x is a positive number, where x>=0.
• The expression under even root should be non-negative. For example the function $\sqrt{\left(x^2+3x-2\right)}$ to be real, $x^2+3x-2\ge 0$ .
• The expression under even root in the denominator of a function should be positive number. For example the function $\frac{1}{\sqrt{\left(x^2+3x-2\right)}}$ to be real, $x^2+3x-2>0$ . Note that zero value of expression is not permitted in the denominator.

Here, we shall work with few examples as illustration for determining domain of real function.

Examples

Problem 1 : A function is given by :

$$f\left(x\right)=\frac{1}{x+1}$$

Determine its domain set.

Solution : The function, in the form of rational expression, needs to be checked for its denominator. The denominator should not evaluate to zero as “a/0” form is undefined. For given function in the question, the denominator evaluates to zero when,