Question 1210395
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What is the inverse of Y=(X+3)² ?
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<pre>
The direct function (which is the given function) does the following:

    - it adds the number 3 to the argument and then squares the value.


Therefore, the range of the given function is the set of all non-negative real numbers.



It means that the domain of the inverse function is the set of all non-negative real  numbers,
and at this set the inverse function makes the opposite operations in reverse order:

    - first, it takes square root from the value, and after that subtract 3 from the square root.


So, we can write an expression for the inverse function

    g(x) = {{{sqrt(x)}}} - 3.



But square root from a real number can be positive or negative.
Therefore, there are TWO possible expressions for "the" inverse function.


First  expression is  p(x) = {{{sqrt(x)}}} - 3,   using positive value of {{{sqrt(x)}}}.


Second expression is  q(x) = -{{{sqrt(x)}}} - 3,  using negative value of {{{sqrt(x)}}}. 


These two possible expressions define two possible inverse functions.


Both inverse functions are defined over the set of non-negative real numbers.


First inverse function has the range [{{{-3}}},{{{infinity}}}).

Second inverse function has the range (-{{{infinity}}},{{{-3}}}].
</pre>

At this point, &nbsp;the problem is solved completely, &nbsp;with detailed explanations, &nbsp;so everybody can understand.



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It is important to note that the question in the problem is not precisely correct/accurate.


Indeed, &nbsp;it asks &nbsp;&nbsp;" what is the inverse function of &nbsp;Y = {{{(X+3)^2}}} &nbsp;?"


To be correct, &nbsp;in this situaion, &nbsp; the question should ask &nbsp;&nbsp;" what are possible inverse functions of &nbsp;Y = {{{(X+3)^2}}} &nbsp;?"



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Now, &nbsp;after reading my post, &nbsp;you are armed to the teeth: 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;you know the right question and also know the right answer.



Come again to this forum soon to learn something new from the best source.