How do you find the inverse of \[y={{\left( x+3 \right)}^{2}}\]?

Hint: This question is from the topic of algebra. In this question, we will first write the equation and then we will replace the variable that is we replace the variable x with y and replace the variable y with x. After that, we will solve the further equation. After that, we will find the inverse of the term \[y={{\left( x+3 \right)}^{2}}\]. Then, we will see the range and domain of \[y={{\left( x+3 \right)}^{2}}\] and it’s inverse.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to find the inverse of \[y={{\left( x+3 \right)}^{2}}\].
The equation which we have to find the inverse is
\[y={{\left( x+3 \right)}^{2}}\]
Now, let us replace x with y and replace y with x. We can write the above equation as
\[x={{\left( y+3 \right)}^{2}}\]
Now, we will find the value of y in terms of x.
Now, squaring root both sides, we can write the above equation as
\[\Rightarrow \pm \sqrt{x}=\sqrt{{{\left( y+3 \right)}^{2}}}\]
The above equation can also be written as
\[\Rightarrow \pm \sqrt{x}=\left( y+3 \right)\]
The above equation can also be written as
\[\Rightarrow \left( y+3 \right)=\pm \sqrt{x}\]
The above equation can also be written as
\[\Rightarrow y=-3\pm \sqrt{x}\]
The above equation can also be written as
\[y=-3+\sqrt{x}\] and \[y=-3-\sqrt{x}\]
So, we have found the inverse of \[y={{\left( x+3 \right)}^{2}}\]. The inverse of \[y={{\left( x+3 \right)}^{2}}\] is \[y=-3\pm \sqrt{x}\].

Note: As we can see that this question is from the topic algebra, so we should have a better knowledge in that topic. Remember that the domain and range of the function will be the range and domain of inverse of the same function. So, we can say that the domain of \[y=-3\pm \sqrt{x}\] will be \[x\ge 0\] and range of \[y=-3\pm \sqrt{x}\] will be \[y\in R\] that is all real numbers. hence, the range of \[y={{\left( x+3 \right)}^{2}}\] will be \[y\ge 0\] and the domain will be \[x\in R\].