Question

How do you find the domain and range of $y = - 2{x^2} + 3$?

Answer 1

We need to find the domain and range of the given function y so the domain refers to the set of possible values of x for which the function y will be defined and the range refers to the possible range of values that the function y can attain for those values of x which are in the domain of the function y.

Complete step-by-step answer:
The given function is $y = - 2{x^2} + 3$.
The function is defined for any value of $x$.
So, the domain of the function is $x \in R$.
As we know the square value of any value is always non-negative.
So, the value of the square is always greater than or equal to zero.
$\Rightarrow {x^2} \ge 0$
Multiply both sides by -1. As we know the inequality reverses on multiplying with -1. So, the above inequality will be,
$\Rightarrow - {x^2} \le 0$
Now add 3 on both sides on the inequality,
$\Rightarrow - {x^2} + 3 \le 0 + 3$
Simplify the terms,
$\Rightarrow - {x^2} + 3 \le 3$
So, the range of the function will never exceed 3.
Hence, the domain of the function is $\left( { - \infty ,\infty } \right)$ and the range of the function is $\left( { - \infty ,3} \right]$.

Note:
Whenever we face such type of problems the key concept is to be clear about the dentitions of domain and range as in this question since we know that ${x^2}$ will be defined for any value of x, extending this concept we can easily find the range of the given function. This understanding of basic definitions will help you get the right answer.
While calculating the range we must consider that in the range of the function always change the interval for which domain is not defined.