Here is why:
When we solve for the inverse function (see the particular step below; when we have: " y² = x " ;
We take the "square root" of EACH SIDE of the equation; to isolate "y" as a single variable on one side of the equation;
→ √(y²) = √x ;
→ |y| = |√x| ;
y = ± √x ;
Because when we take the square root — or any "even root", for that matter—we have two solutions: of a variable, we have TWO SOLUTIONS: a positive value; and a negative values;
→ since: 1) a "negative value"; multiplied by a "negative value" ; equals a "positive value" ;
→ and as such: a "negative value" ;
multiplied by that same "negative value" ;
{that is: a "negative value", "squared (i.e "raised to the power of "2"} ; ,
→ results in a positive value ;
→ and since:
2) a "postive value"; multiplied by a positive value" ; equals a "positive value" ;
→ and as such: a "positive value" ;
multiplied by that same "positive value" ;
{that is: a "positive value", "squared (i.e "raised to the power of "2"} ; ,
→ results in a "positive value";
→ and since:
3) any given integer, in it "positive value", squared (i.e. raised to the power of "2"); results in a "positive value" ;
→and since:
4) that same aforementioned integer; in its "negative value" form, squared (i.e. raised to the power of "2"); results in that same aforementioned "positive value".
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Note the following:
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"Given the function: "f(x) = x² " ; Find the "inverse function" .
Let "y" = f(x) " ;
and write as: " f(x) = y = x² " ;
→ " y = x² " ;
→ Now, rewrite the equation; replacing the "y" with "x" ;
and replacing the "x" with "y" ;
→ " x = y² " ;
Now, rewrite the question; isolating "y" as a single variable;
with no coefficient (save for the "implied coefficient of "1" ) ;
→ " x = y² " ;
↔ " y² = x ;
Now, take the square root of EACH SIDE of the equation;
to isolate "y" on one side of the equation;
→ √(y²) = √x ;
→ |y| = |√x| ;
→ y = ± √x .
Replace the "y" with " f ⁻¹(x)" ; to indicate that this the "inverse function" ;
and write the "inverse function" :
→ " f ⁻¹(x) = ± √x " ;
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