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Find the range of \[f\left( x \right) = \sqrt {9 - {x^2}} \], range is \[\left[ {a,b} \right]\] , then \[b - a = ?\]

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Last updated date: 22nd Jun 2024
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Answer
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Hint:
Here we need to find the range of the given function. Range of a function is defined as the set of values which are given out from the function by substituting the value of the domain of the function. Here, the given function is the square root of some function. So the expression inside the square root can’t be less than zero but can be equal or greater than zero. We will use this condition to find the value of the variable and hence the range of the given function.

Complete step by step solution:
The given function is \[f\left( x \right) = \sqrt {9 - {x^2}} \].
Here, we need to find the range of this function.
We know that the terms inside the square root can’t be zero but can be equal or greater than zero.
Now, we will write the condition mathematically.
\[ \Rightarrow 9 - {x^2} \ge 0\]
Now, subtracting 9 on both sides, we get
\[\begin{array}{l} \Rightarrow 9 - {x^2} - 9 \ge - 9\\ \Rightarrow - {x^2} \ge - 9\end{array}\]
Multiplying \[ - 1\] on both sides, we get
\[ \Rightarrow {x^2} \le 9\]
Taking square roots on both sides, we get
\[ \Rightarrow \left| x \right| \le 3\]
We can write this inequality as
\[ \Rightarrow - 3 \le x \le 3\]
Therefore, we can say that
\[x \in \left[ { - 3,3} \right]\]
Now substituting \[x = 0\] in \[f\left( x \right) = \sqrt {9 - {x^2}} \], we get
\[ \Rightarrow f\left( x \right) = \sqrt {9 - {0^2}} = \sqrt 9 = 3\]
Now substituting \[x = 3\] in \[f\left( x \right) = \sqrt {9 - {x^2}} \], we get
\[ \Rightarrow f\left( x \right) = \sqrt {9 - {3^2}} = \sqrt {9 - 9} = 0\]
Hence, the range of the function is \[\left[ {0,3} \right]\]
Given range of the function is \[\left[ {a,b} \right]\].
On comparing these ranges, we get
\[\begin{array}{l}a = 0\\b = 3\end{array}\]
Here, I need to calculate \[b - a\].
On substituting the obtained values, we get
\[ \Rightarrow b - a = 3 - 0 = 3\]

Note:
1) We have solved the given inequality to find the value of the variable \[x\]. If the expression on the left-hand side and right-hand sides are unequal, that would be called an inequality equation. We need to know some basic and important properties of inequality. Some important properties of inequality are as follows:-
2) If we add or subtract a term from both sides of an inequality, then inequality remains the same.
3) If we multiply or divide a positive number on both sides of an inequality, then inequality remains the same.
4) But if we multiply or divide a negative number on both sides of an inequality, then inequality does not remain the same.