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# Function

A function is a rule which associates two or more than two variables. Functions can be understood with the help of some very basic examples:
1. Area of a circle (A) = πr2, where r is the radius.

So, the area of a circle is dependent upon the value of the radius of a circle. We can write this mathematically as A = f(r)
2. If the distance (d) is constant, then the time taken (t) to cover that distance (d) will be dependent on the value of the speed (v). This can be written mathematically as t = f(v)

Normally any function is represented in the format of y = f(x)
Domain and range of y = f(x) In case of y = f(x), values of y are dependent upon the corresponding values of x. Here y is known as the dependent variable and x is known as the independent variable.

All the real values of x for which atleast one real value of y exists, are known as the domain of this function y = f(x)

All the possible real values of y are known as the range of this function y = f(x).

While finding out the domain, the variable generally takes a range of values unlike the case of equations where the variable will, mostly, take one value or a discrete set of values in interval notation. Hence, it is important to understand the standard notations which are used to represent the solution set of the variable as well as the interval notation.
• (a, b) read as “open interval a, b” means all real numbers between a and b excluding a and ba < b.
• [ab] read as “closed interval ab” means all real numbers between a and b including a and ba < b.
• [ab) means all numbers between a and b, with a being included and b excluded; a < b.
• (ab] means all numbers between a and b, with a being excluded and b included; a < b.
Example-1
Represent all real numbers between 1 and 10 in the interval from where
1. 1 and 10 are included,
2. 1 and 10 are excluded
3. 1 is included and 10 is excluded and
4. 1 is excluded and 10 is included
Solution
1. [1, 10] i.e., 1 ≤ x ≤ 10 and x is a real number.
2. (1, 10) i.e., 1 < x < 10 and x is a real number.
3. [1, 10) i.e., 1 ≤ x < 10 and x is a real number.
4. [1, 7] i.e., 1 ≤ x ≤ 7 and x is a real number.

Process to find out the domain of y = f(x)

1. Denominator should not be equal to zero.
2. Expression under square root should be a non-negative.

Example-2
Find the domain of the definition of function y = f(x) = .
Solution
We know that the expression under square root should be non-negative.

So, 4 – x ≥ 0, or, x ≤ 4
And x – 6 ≥ 0, or, x ≥ 6.

Arranging these values on the numberline:

So, no value of x is common. Hence the domain of y = f(x) is not defined.

Example-3
The domain of y = f(x) =  is
Solution
For y = f(x) to be defined, |x| −x>0, or, |x|>x.

It is true only if x < 0. So, the domain is −∝ < x < 0.

Example-4
Are the following functions identical?
y = f(x) = x

y = h(x) =

Solution
Domain of f(x) = all the real values

Domain of h(x) = all the real values except at x = 0.

Because the domains are not the same for f(x) and h(x), so f(x) and h(x) are not identical.