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Functions in math are just like a vending machine. We give money as input to the vending machine and get some chocolates or cookies in return as the output. The money that we put in the vending machine is the domain of the machine.
Thus, the meaning of domain is the set of all the possible inputs for which the function works and gives real values as output. The basic process can be represented pictorially as below:
For example, for a function f(x)=x-9 , the domain will be x≥9 because we know that the square root can never be negative. Therefore, x-9≥0 ⇒ x≥9 . The function does not output real values for x<9 . Thus, the domain will be x≥9 .
HOW TO FIND THE DOMAIN OF A FUNCTION?
To find the domain, we have to find all the values of the independent variables which are allowed to be used. We have to be careful of those values of the independent variable which makes the function undefined.
For example, the denominator of a fraction can never be zero because it makes the function undefined. Also, there cannot be a negative sign inside the square root.
Graphical method:
We can find the domain of a function using a graph. It is the most effective way to find the range of any function.
To find range using a graph, follow the below steps:
Draw the graph of the given function.
Note down the minimum and maximum value of the variable x on the x-axis over which the graph is spread.
The domain of the function will be [minimum x value, maximum x value]
For example, Let us find the domain of the function f(x)=x-9 using a graphical method.
We know that the graph of f(x)=x-9 is as below:
From the graph, we can see that the minimum value of x for which the graph is defined is x=9 and the maximum value is infinity.
Therefore, the domain of the function will be [9,).
The domain of some important functions :
NOTE: In the above table, ‘R’ means ‘All Real numbers’ and ‘a’, ‘b’, ‘c’ are constants and ‘n’ means integers.
The bracket ‘[‘ and ‘]’ indicates that the number is included in the domain.
The bracket ‘(’ and ‘)’ indicates that the number is excluded from the domain.
EXAMPLES:
Find the domain of f(x)=6xx2-9
To find the domain of functions including fractions, we must ensure that the denominator never becomes zero. Any function having zero in the denominator makes the function undefined.
Here, let us calculate the values of x for which the denominator becomes zero.
x2-9=0
(x-3)(x+3)=0
⇒ x=3 and x=-3
Thus, the values of x for which the denominator becomes zero are x=3 and x=-3.
These values of x make the function undefined. Therefore, these values will be excluded from the domain.
Hence, the domain of the given function is “All Real Numbers except x=3,-3 ”.
Find the domain of f(x)=log(x-5)
From the table of domains discussed above, we know that the logarithmic function takes only the positive values as a domain.
Thus, we need to set the terms inside the log greater than zero.
x-5>0
x>5
Therefore, the domain of the given function will be (5,∞).
Conclusion–
A domain is an important parameter that defines a function. One must know how to find the domain of a function to know how the function behaves and what are its limits. We have discussed two methods to find the domain of a function. It is always easier and more effective to find the domain of a function using the graphical method. But one must have the knowledge of graphs of the functions one is dealing with.
