Differential Calculus-Concept of a Real-Valued Function

Introduction

We can say that the domain of a function represents the values needed in the function for the function to exist independently of real numbers. As we talk about real numbers, we must avoid division by zero and logarithms of zero. We should avoid square roots of negative numbers and their logarithms too.

For example, suppose S⊆N be a subset of a set of real numbers, i.e., N. Now assume that a corresponding value of the dependent variable y exists for each value of the independent variable z of S. So, we can conclude that y (dependent variable) is a real function of z (the independent variable).

The graph of a real-valued fraction is a Cartesian period. Concept of a Real-Valued Function – Domain, Range and Graph of a Function is an important topic in the calculus of mathematics.

Domain and range

Domain and range are always defined for a relation. They are the sets of all the x and y-coordinates of ordered pairs. To simplify, we can say that a domain shall be regarded as the set with all the possible outcomes or x-values due to which the function will work with real y-output values.

On the other hand, the range is the entire set of all possible outcomes or values of any dependent variable, say y, after substitution of the domain.

As already discussed, A real-valued function is a type of function whose domain is always a subset D Í N of the set N having only real numbers, and R is also a codomain. On the other hand, the codomain is a subset of R (real numbers). Therefore, the range of a real-valued function will be the set of points such that y = f(x).

Important points to remember

• There should never be zero in the denominator.
• If a square root is present, then the number under it must be a positive square. 

How to determine a domain

We determine the domain of a function by identifying the independent variables, i.e., mostly x.

How to determine a range

On the other hand, the range is all possible outputs of y-values. These are the minimum to maximum values of y.

For example:

Suppose a relation R is given as:

R={(1,3) ,(2,2), (3,2), (4, 3)},then the domain and range of this relation shall be as follows:

Domain = {1, 2, 3, 4} [the set of all x-coordinates]

Range = {2, 3} [the set of all y-coordinates]

• Domain and Range of Exponential Function: The domain and range of an exponential function is always a real line and positive values.
• Domain and Range of an Absolute Value Function: The domain of an absolute value function is R, whereas its range is [0, ∞].

Solved Question on Concept of a Real-Valued Function – Domain, Range and Graph of a Function

Q: A function is given as (x+1)/(3-x). find its domain and range?

Solution: Setting the denominator equals to zero to solve for x.

We get: 3-x=0

Or, – x = – 3 

x = 3

Excluding 3 from the domain (as x =3)

Now, we know that the domain is always a set of real numbers. 

= (x< 3)

And (x > 3)

Range of y, y=(x+1)/(3-x)

Solving for equation x,

x + 1 =(3 – x)y

x + 1=3y – xy

x + xy=3y-1

3y – 1= x(1 + y)

x = (3y – 1)/(1 + y)

As this final equation, i.e., a fraction is not defined (if the denominator is zero), therefore

1+y ≠ 0 

⇒ y≠ -1

So, we may conclude that the range of this function consists of all sets of natural numbers (excluding -1).

Answer: Domain of the given function = (-∞, 3) ∪∪ (3, ∞)

Range = (-∞,-1) ∪∪ (-1, ∞)

Conclusion

A real-valued function is a type of function whose domain is always a subset D Í N of the set N having only real numbers, and N is also a codomain. As we talk about real numbers, we must avoid division by zero and logarithms of zero. We should avoid square roots of negative numbers and their logarithms too.

The domain shall be regarded as the set with all the possible outcomes or x-values due to which the function will work with real y-output values. On the other hand, the range is the entire set of all possible outcomes or values of any dependent variable, say y, after substitution of the domain. We determine the domain of a function by identifying the independent variables, i.e., mostly x.