Question

What is the domain and range of $y = - \sqrt {9 - {x^2}} $ ?

Answer 1

Hint: To simplify this question , we need to solve it step by step . Here we are going to determine the domain of $y = f(x)$ , find out the set of values that the variable $x$ in the function can have and to determine the range express the function as x in terms of $y$ and consider the fact that the range $y = f(x)$of will be the domain of $x$.

Complete step-by-step solution:
Given a function in the above question, $y = - \sqrt {9 - {x^2}} $
$f(x) = - \sqrt {9 - {x^2}} $
Domain is basically the set to values that the x in the function can have.
The only restriction that we have is that for real numbers, you can only take the square root of a positive number.
In other words, in order for the function to be defined, we need the expression that shoulb be under the square root and that to be positive.
Since, $\sqrt {9 - {x^2}} $is always greater than zero for all values of x.
 $
  \sqrt {9 - {x^2}} \geqslant 0 \\
  {x^2} \leqslant 9 \Rightarrow |x| \leqslant 3 \\
 $
This means that we are having ,
$
  x \geqslant - 3 \\
  x \leqslant 3 \\
 $
For any value of x outside the interval \[[ - 3,3]\] , the expression under the square root will be negative, which means that the function will be undefined. Therefore, the domain of the function will be  \[x \in [ - 3,3]\].

Now If we talk about the range. For any value of \[x \in [ - 3,3]\] , the function will be negative.

The maximum value of the expression under the radical can take is for $x = 0$

\[9 - {0^2} = 9\] which means that the minimum value of the function will be

\[y = - \sqrt 9 = - 3\] . Therefore, the range of the function will be \[[ - 3,0]\].

Therefore, the domain of $f(x) = - \sqrt {9 - {x^2}} $ is \[x \in [ - 3,3]\] and range is \[[ - 3,0]\].
Additional Information:
DOMAIN: Let R be a relation from a set A to a set B. Then the set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R.
Thus, domain of $R= \{ a:\left(a,b\right)\in R \}$
Clearly, the domain of $R \subseteq A$.
If A={1,3,5,7}, B={2,4,6,8,10} and R={(1,8),(3,6),(5,2),(1,4)} is a relation from A to B,
Then,
Domain R={1,3,,5}
RANGE: Let R be a relation from a set A to a set B . Then the set of all second components or coordinates of the ordered pairs belonging to R is called the range of R.
Thus, Range of $R= \{b:\left(a,b\right)\in R \}$
Clearly, range of $R \subseteq B$
If A={1,3,5,7}, B={2,4,6,8,10} and R={(1,8),(3,6),(5,2),(1,4)} is a relation from A to B,
Then,
Range R={8,6,2,4}

Note: Always try to understand the mathematical statement carefully and keep things distinct. Remember the properties and apply appropriately. Don’t forget to cross-check your answer at least once.
$R$ represents the set of all real numbers.
$R_ + ^*$ represents the set of all positive real numbers.
Choose the options wisely , it's better to break the question and then solve part by part .
Cross check the answer and always keep the final answer simplified .