Question

Find the domain and range of the following functions: a) f(x) = under root 16-x^2 b) f(x) = x^2/1+x^2 ; x belongs to R

Answer 1

Consider the first function to calculate its domain and range.

1. $\sqrt{16-x^{2}}$

Domain of a function is the set of input or argument values for which the function is defined.

This function is defined for the values of 'x' which are greater than -4 but less than 4.

As if we take 'x' as 5 or -5, we get $\sqrt{-9}$, which is not defined.

So, the domain of this function = ${x\epsilon \Re : -4\leq x\leq 4}$

Now, we will find the range of this function.

The range of a function is the complete set of all possible resulting values of 'y', after we have substituted the domain.

So, after substituting the domain values in the given function, we get the values of 'y' between 0 and 4.

So, We get range as = ${y\epsilon \Re : 0\leq y\leq 4}$

2. Now consider the second function $\frac{x^{2}}{1+x^{2}}$

Domain of a function is the set of input or argument values for which the function is defined.

We observe that the given function is defined for the values of 'x'.

So, the domain of this function = ${x\epsilon \Re}$

Now, we will find the range of this function.

The range of a function is the complete set of all possible resulting values of 'y', after we have substituted the domain.

So, after substituting the domain values in the given function, we get the values of 'y' between 0 and 0 and 1.

So, We get range as = ${y\epsilon \Re : 0\leq y< 4}$