Domain and range  (Page 4/4)

1. Total expression within the square root as a whole is non-negative number as square root of a negative number is not a real number.
2. For positive value of "y" in the numerator, the denominator is non-negative as square root of a negative number is not a real number.
3. The denominator does not evaluate to zero. The form “y/0” is undefined.

For the first requirement, the expression in the square root should be greater than or equal to zero i.e non-negative number.

$$\frac{y}{1-y}\ge 0$$

$$\Rightarrow y\ge 0$$

Further, denominator of the expression “1-y” is non-negative. Also, “1-y” is not equal to zero. Combining two requirements, the expression is a positive number :

$$1-y>0$$

$$\Rightarrow y<1$$

Combining two intervals i.e. intersection of two intervals, we have the range of the function as :

Range of a function

$$\Rightarrow 0\le y<1$$

Classification of functions

In mathematics, we deal with specific real functions, which are characterized by specific domain, range and rules. Some of the familiar functions are polynomial, rational, irrational, trigonometric, exponential, logarithmic functions and piece wise defined functions etc. These functions are further combined to form more complex function following certain definition or rule so that function is meaningful for real values.

There are varieties of functions. These functions are broadly classified under three headings :

• Algebraic functions : polynomial, rational and irrational functions
• Transcendental functions : trigonometric, inverse trigonometric, exponential and logarithmic functions
• Piece wise defined functions : modulus, greatest integer, least integer, fraction part functions and other specific piece wise definitions

Polynomial function is further classified based on (i) numbers of terms eg. monomial, binomial, trinomial etc. (ii) numbers of variables involved eg. function in one or two variables and (iii) degree of the polynomial eg. linear, quadratic, cubic, bi-quadratic etc.

Generally we deal with function expressions in one variable in which dependent variable (y) is explicitly related to independent variable (x). Such functions are called explicit function.

$$y=x^2-2x+1$$

On the other hand, there are function rules in which “y” is not explicitly related to “x”. Such functions are called implicit functions.

$$\sin \left(x^2+xy+y^2\right)=xy$$

Further, we also use properties of function like periodicity (repetition of function values at regular intervals of independent variable) and polarity (odd or even) to characterize a function. For this reason, we sometimes name a function like periodic, non-periodic, aperiodic, odd, even or equal function etc.

Important concepts

In this section, we discuss few important concepts, which are frequently used in determining domain and range.

Defined and undefined expressions

Defined expressions are meaningful and unambiguous. On the contrary, undefined expressions are not meaningful. Most of the undefined expressions results from combination of zeros and infinity in various ways. There is, however, no unanimity about “undefined” values among mathematicians. Hence, we shall present two lists – one list which is undefined in all contexts and another list which may be defined in certain context. We consider this later list as defined expressions, unless otherwise stated.

Undefined in all contexts

$$\frac{x}{0},\infty -\infty ,{\left(-1\right)}^{\infty },\frac{\infty }{\infty },0X(-\infty )$$

We should understand here that an expression is undefined when it can not be interpreted. The important point is that it has got nothing to do with the magnitude of quantity. Emphatically, infinity is not undefined. We shall discuss this aspect subsequently.

Note that " ${\left(-1\right)}^{\infty }$ " is undefined, because it is not certain whether the expression will evaluate to "-1" or "1". On the other hand, expression " ${\left(1\right)}^{\infty }$ " is defined because we are sure that the expression will evaluate to "1", however large is infinity.

Defined in some contexts

$$0^0=\text{undefined or 1}$$

$${\infty }^0=\text{undefined or 1}$$

$$1^{\infty }=\text{undefined or 1}$$

$$0X\infty =\text{undefined or 0}$$

For our purpose, unless otherwise stated, we shall consider later set as defined.

Infinity

Infinity is not a member of real number set “R”. Strictly we can not write infinity like :

$$x=\infty ,\text{where “x” is real.}$$

For this reason, the interval of real number set is defined in terms of infinity without equality as :

$$-\infty <x<\infty \text{or}\left(-\infty ,\infty \right)$$

We may emphasize that infinity by itself is “unbounded” – not undefined. What it means that we can interpret infinity – even though its value is not known. We can say it is very large number (either positive or negative as the case be), but we can not interpret undefined values at all.

It is easy to interpret operations with infinity. We need to only keep the meaning of infinity as a very large number in focus. Various operations, involving infinity are presented here :

1: The plus or minus infinity is not changed by adding or subtracting real number.

$$\infty \pm x=\infty$$

$$-\infty \pm x=-\infty$$

Above results are on expected line. Addition or subtraction of finite values will only yield a large number. It is so because infinity can be greater than a large value that we might conceive.

2: Addition of two infinities is infinity.

$$\infty +\infty =\infty$$

3: Difference of two infinities is undefined.

$$\infty -\infty =\text{Undefined}$$

Addition of two infinities is definitely a very large number. We are, however, not sure about their difference. The difference of two infinities can either be small or large. It depends on the relative "largeness" of two infinities. Hence, difference of two infinities is "undefined".

4: Product of two infinities are inferred as :

$$\infty X\infty =\infty$$

$$-\infty X\infty =-\infty$$

$$-\infty X-\infty =\infty$$

5: Product of infinity with a real number “x” is given as :

$$xX\infty =\infty ,\text{if}x>0$$

$$xX\infty =-\infty ,\text{if}x<0$$

$$xX\infty =\mathrm{0,}\text{if}x=0$$

6: Division of infinity by infinity is not defined.

$$\frac{\infty }{\infty }=\text{Undefined}$$

7: A real number, “x”, raised to infinity

$$x^{\infty }=\mathrm{0,}\text{if}0<x<1$$

$$x^{\infty }=\mathrm{0,}\text{if}x=0$$

$$x^{\infty }=\infty ,\text{if}x>1$$

8: An infinity raised to infinity is defined.

$${\infty }^{\infty }=\infty$$