Question

How do you find the domain and range of $f\left( x \right) = 10 - {x^2}$?

Answer 1

Here we must know that domain is the value of all the variables that can be placed in place of it. The range is the value that we can get of the function after substituting all the values of the domain. So here we can see that in the above function given $f\left( x \right) = 10 - {x^2}$ the $x$ can take any value but the range here cannot be more than $10$ as the square of $x$ is always positive.

Complete step by step solution:
Here we are given the function $f\left( x \right) = 10 - {x^2}$ of which we need to find the domain and range. For this, we must know what the meaning of the domain and range of the function is. The domain of any function is just the value that can be placed in the place of the variable. The value of the function that is obtained after substituting the value of all the variables that come under the domain is called the range. If the function is contained inside the root-like we have $\sqrt {x - 1}$ then we know that here any number inside the root can never be less than $0$ hence the domain will be $x - 1 \geqslant 0$.
Similarly here we are given the function $f\left( x \right) = 10 - {x^2}$
So here we can substitute any value of $x$
Hence domain$= \left( { - \infty ,\infty } \right)$
Now we can find the value of range by finding the value of the function that comes under a domain.
So we have $f\left( x \right) = 10 - {x^2}$
So we know that ${x^2} \geqslant 0$
Hence now with the negative sign we will get this as:
$- {x^2} \leqslant 0$
Adding $10$ to both the sides we will get:
$10 - {x^2} \leqslant 10 + 0$
Hence we will get:
$10 - {x^2} \leqslant 10$

Hence we can say that range$= \left( { - \infty ,10} \right]$

Note:
Here the student must know that when we have the number inside the square root then we need to just keep the value under the root greater than or equal to $0$ and also while finding the range we can actually find the domain of the $x$.