Question

What is the domain and range of \[y={{x}^{2}}+1\] ?

Answer 1

Hint: These type of problems are quite simple in nature and are very easy to solve. Questions like these can be solved very quickly once we understand the underlying concepts behind the problem. To solve this particular problem efficiently, we need to have some basic as well as advanced knowledge of chapters like graph theory, functions and differential calculus. In such cases to find the domain of the function we need to find all the possible values of ‘x’ that satisfies the equation and doesn’t lead to any undefined value of ‘y’. On the other hand, to find the range of the function, we need to find all the possible values of ‘y’ for all the corresponding values of ‘x’.

Complete step-by-step answer:
Now we start off with the solution to the given problem by writing that, for all real values of ‘x’ there is a corresponding real value of ‘y’. There is no such value of ‘x’ such that the value of ‘y’ becomes undefined. So from this observation we can clearly say that the domain of the given function is \[x\in \mathbb{R}\] .
Now coming to the range of the function, we can clearly observe that the value of ‘y’ can never be negative because of the presence of the term \[{{x}^{2}}\] in the right hand side of the equation. We can also see that the minimum value of the function ‘y’ is equal to \[1\] which comes at \[x=0\] . So we can say that the range of the function is, all positive values excluding the interval zero to \[1\] . This range is \[y\in {{\mathbb{R}}^{+}}-\left( 0,1 \right)\] .

Note: For such problems we need to be thorough with chapters like graphs and functions. This particular problem can also be solved using graphs. In this method we plot the graph for the given function and then analysing from it, we can clearly find the domain and range of the given function. We need to be very careful with the points which lead to an undefined value and asymptotes or else may cause an error in the problem leading to a wrong solution.