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Question: What is the domain and range of \(y = - \sqrt {9 - {x^2}} \) ?...

What is the domain and range of y=−9−x2y = - \sqrt {9 - {x^2}} ?

Explanation

Solution

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

Complete step-by-step solution:
Given a function in the above question, y=−9−x2y = - \sqrt {9 - {x^2}}
f(x)=−9−x2f(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, 9−x2\sqrt {9 - {x^2}} is always greater than zero for all values of x.
9−x2⩾0 x2⩽9⇒∣x∣⩽3  \sqrt {9 - {x^2}} \geqslant 0 \\\ {x^2} \leqslant 9 \Rightarrow |x| \leqslant 3 \\\
This means that we are having ,
x⩾−3 x⩽3  x \geqslant - 3 \\\ x \leqslant 3 \\\
For any value of x outside the interval [−3,3][ - 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∈[−3,3]x \in [ - 3,3].

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

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

9−02=99 - {0^2} = 9 which means that the minimum value of the function will be

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

Therefore, the domain of f(x)=−9−x2f(x) = - \sqrt {9 - {x^2}} is x∈[−3,3]x \in [ - 3,3] and range is [−3,0][ - 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:(a,b)∈RR= \\{ a:\left(a,b\right)\in R \\}
Clearly, the domain of R⊆AR \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:(a,b)∈RR= \\{b:\left(a,b\right)\in R \\}
Clearly, range of R⊆BR \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.
RR represents the set of all real numbers.
R+∗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 .