Question

How do you find the domain and range for \[y={{\left( x-5 \right)}^{2}}\]?

Answer 1

Hint: For the question we are asked to find the domain and range of the function \[y={{\left( x-5 \right)}^{2}}\]. So, for this type of questions we will find the values of x where the y is valid then those x values will be domain and we will find the range of the function by substituting the domain of the function then the values of y which we will get are the range.

Complete step by step solution:
Here the function is defined for all the real numbers. So, the function has values that is it has y values for all the x which are real numbers.
So, the domain of the function is,
\[\Rightarrow x\in R\]
If we carefully observe the given function we can say that it is a square function. So, it is always positive for any value of x which is in the domain.
So, the function value can go from \[0\] to \[\infty \].
Therefore, the range of the function simply the complete set of all possible values of y which we get when we substitute the domain of the function in the function is,
\[\Rightarrow R=[0,\infty )\]
Therefore, for the given question \[y={{\left( x-5 \right)}^{2}}\] the domain and range will be \[x\in R\] and \[R=[0,\infty )\] respectively.

Note:
Students must be having good theoretical knowledge in basic definitions of the domain and range of the function and the concept of functions.
Here students must be very careful that the range of this kind of square function will be having \[[0,\infty )\] square bracket to zero as the function will attain value zero when the \[x=5\].
\[\Rightarrow y={{\left( x-5 \right)}^{2}}\]
\[\Rightarrow y={{\left( 5-5 \right)}^{2}}\]
\[\Rightarrow y=0\]
So, students must be very careful in this regard.