Question

Find range and domain of the following \[\sqrt{4-{{x}^{2}}}?\]

Answer 1

Hint: Use the definition of domain and find the domain of the function. Similarly use the definition of range and find the range of the function. For solution of inequality use the concept of open and close intervals and write the domain and range in standard form.

Complete step-by-step answer:
Domain: Domain is the set of all real numbers \[x\] for which \[f\left( x \right)\] is a real number.
Range: Range of real function is the set of all values of \[f\left( x \right)\] such that \[x\] belongs to the domain of the function.
If \[y=f\left( x \right)\]
Where
Value of \[x\]is known Domain of \[f\left( x \right)\]
\[y=\sqrt{4-{{x}^{2}}}\]
Condition for the domain
\[\Rightarrow 4-{{x}^{2}}\ge 0\]
Simplify the equation
\[\Rightarrow (2-x)(2+x)\ge 0\]
\[Take\,\,(-)\]sign common
\[\Rightarrow (x-2)(x+2)\le 0\]
Use the concept of the number line system
Domain of the function
\[x\in [2,2]\]
Now, solve for range of \[f\left( x \right)\]
\[y=\sqrt{4-{{x}^{2}}}\]
Squaring both sides of the equation, we get
\[
  {{\text{(y)}}^{\text{2}}}{\text{ = (}}\sqrt {{\text{4 - }}{{\text{x}}^{\text{2}}}} {{\text{)}}^{\text{2}}} \\
   \Rightarrow {y^2} = 4 - {x^2} \\
  \therefore {x^2} = 4 - {y^2} \\
\]
Simplify the equation by taking root both sides
\[\Rightarrow x=\sqrt{4-{{y}^{2}}}\]
From this expression we can say the value of \[y\] will be greater than or equal to \[0.\]
\[\Rightarrow y\ge 0\]
And the maximum value of \[y\] will be less than or equal to \[2.\] so that the value inside the square root will be positive for real value of \[x\]
\[\Rightarrow y\le 2\]
That means the range of the function will be
\[\Rightarrow y\in [0,2]\]

Hence, the domain of the function will be \[x\in [-2,2]\] and range of the function will be \[y\in [0,2]\]

Note: The problem can also be solved by the method of graphing. In this method we draw the graph and find the values of domain and range from the values on x-axis and y-axis.
This type of expression is always solved by the concept of inequality. Solutions of inequalities are always written in the form of intervals like close and open intervals.
Close bracket is applied when the value is greater than equal to or less than equal to and open bracket is for the value is greater than or less than that value.