Question

Find the domain and range of \[f\left( x \right) = \sqrt {x - 5} \]

Answer 1

Hint: Here the function is in root so first we will find the value of domain then from that we can get the value of range. We must remember that the square root can never be negative.

Complete step-by-step answer:
Given function
\[y = f\left( x \right)\]
\[y\]\[ = \]\[\sqrt {x - 5} \]\[ \to \]\[\left( 1 \right)\]
To find domain and range of the given function \[f\left( x \right) = \sqrt {x - 5} \]
Now , we have first calculate domain of the given function
Therefore , The domain of \[f\left( x \right) = \sqrt {x - 5} \] is the set of all real \[x\] for which \[f\left( x \right)\] is meaningful or defined
Rules for finding domain
\[ \Rightarrow \] Expression under even roots ( square roots or fourth roots ….) \[ \geqslant \] 0
\[ \Rightarrow \] Denominator is not equal to zero
Now ,\[f\left( x \right)\] to be defined the term under the square – root should be greater than or equal to zero
So ,\[x - 5 \geqslant 0\]
\[x \geqslant 5\]
Therefore , domain of the given function is \[\left[ {5,\infty } \right)\]
Again , we have calculate range of the given function \[f\left( x \right) = \sqrt {x - 5} \]
Therefore , range of \[f\left( x \right) = \sqrt {x - 5} \] is the collection of all outputs of \[f\left( x \right)\] corresponding to each real numbers in domain
Rules for finding the range
\[ \Rightarrow \] If domain \[ \in \]set of finite number of points implies range \[ \in \]set of corresponding \[f\left( x \right)\] values
\[ \Rightarrow \] If domain \[ \in \]R or R – ( set of finite points ) then \[x\]is expressed in terms of \[y\]
Now ,\[f\left( x \right)\] to be defined the term under the square- root should be greater than or equal to zero
So ,\[x - 5 \geqslant 0\] \[ \to \]\[\left( 2 \right)\]
Square root of equation \[\left( 2 \right)\], we get
\[\sqrt {x - 5} \]\[ \geqslant 0\]
\[y \geqslant 0\]
Since , range of the given function is \[\left[ {0,\infty } \right)\]

Therefore , Domain of the function \[f\left( x \right) = \sqrt {x - 5} \] is \[\left[ {5,\infty } \right)\]
And Range of the function \[f\left( x \right) = \sqrt {x - 5} \] is \[\left[ {0,\infty } \right)\]

Note: The domain of a function f(x) is defined as the set of all values for which the function is defined, and the range is defined as the function is the set of all values that f takes. (In grammar school, you probably called the domain the replacement set and the range the solution set.