Question

How do you find the range of a function?

Answer 1

Now first let us understand the concept of a function.
A function is nothing but takes in a value and gives another value. Hence a function takes an input and gives an output.
For example let us take $f\left( x \right)=x+2$ .
Now if we substitute x = 2 in the above function we get, $f\left( x \right)=4$
Hence we say that 2 is input of the function and 4 is output of the function.

Complete step by step answer:
Now let us understand the concept of domain and range.
Domain of a function is all the set of values which a function can accept as input. Similarly range is all the set of values which a function can give as output.
Now let us check this by an example.
Consider the function $f\left( x \right)=\left| x \right|$
Now here we can see that the function can accept all the values as input. Hence the domain of the function is the set of all real numbers.
But now for any input we know that the value of the function will always be positive or zero.
Hence the function has a range as a set of all non-negative numbers.
Similarly consider the function $\sqrt{x}$
Now we know that the root of a negative number is not defined.
Hence we cannot take negative numbers as input for the function.
Hence we have the domain of the function is all non-negative numbers.
Similarly we can say that the range of the function is all non-negative numbers as the square root of any number is never negative.
Now in general we have functions expressed as y in terms of x.
To find the range of the function we will convert the function such that we get x in terms of y. Then we find the domain of the new function obtained. This will be the range of the original function.
For example consider the function $y={{x}^{2}}$ .
Now to find the range of the function we will write x in terms of y.
Hence we get, $x=\sqrt{y}$ .
Now we know that the domain of the above function is a set of all non-negative numbers.
Hence the range of $y={{x}^{2}}$ is set of all non-negative numbers.

Note: Now note that when we say set of non-negative numbers we mean the set of all positive numbers and 0. Also note that when checking the domain we have to follow general rules of Mathematics like the root of a negative number is not defined or the denominator of a fraction can never be zero etc.