Question

How do you find the domain and range of $y = {x^2} + 1?$

Answer 1

As we know that the domain are numbers that give us the function. And range refers to the numbers that the function gives back. In this question we need to find the domain and range of the function, we know that domain refers to all the possible values of $x$ for which the function will be defined.

Complete step by step solution:
According to the question we have the function $y = {x^2} + 1$. We can see that any real value can be the possible input value of $x$,
So we can say that Domain is $x \in R$ or $x[\left( { - \infty ,\infty } \right)$.
Now we know the general form of the quadratic equation with vertex i.e.$f(x) = a{(x - h)^2} + k$, where $h,k$ are the vertex.
Similarly we can write the given equation as$y = {(x - 0)^2} + 1$. By comparing both the equations we have $h = 0,k = 1$ and $a = 1$. So the vertex is $(0,1)$.
As we know that if $a$ is positive then the parabola opens upward and here the value of $a$ is positive.
We can see that the vertex is at the minimum point so $x = 0,y = 1$. It is less than and equal to one.
So we can write that the range is $y \geqslant 1$ or $y[1,\infty )$.
Hence the domain is $x \in R$ or $x[\left( { - \infty ,\infty } \right)$ and the range is $y \geqslant 1$ or $y[1,\infty )$.

Note: Before solving this kind of question we should have the proper knowledge of domain, range and their functions. Whenever we have such type of questions the key concept is to be clear about the definitions of domain and range. We should remember that if the expression is given in fraction then the domain of a fraction cannot be zero.