absolute value of x (|x|)
If you graph the square root of x2 using any graphing calculator
you'll see the graph of y = |x| and not y = x.
The square root of a number x is defined such that a2 = x,
where a is the square root of x. Then you might say the square root of x
2 should be x and not |x| because according to the defination
of square root function x2 = x2, doesn't matter if
x is negative because negative2 will be a positive number.
All that is right except that the defination of square root was not complete. A function is a relation that associates an input with a single output, any function in its domain cannot give two outputs for a single input. This is why we restrict the domain while working with inverse trigonometric functions.
If we define square root of x2 as x instead of |x|, we will break the number line and get wrong results. For example, square root of (-2)2 should be 2 but let's say its -2. Now simplify the left hand side of the equation to square root of 22, since (-1) 2 = 1. From above two equations we get 2 = -2 which is wrong! If we define the square root of x2 as |x| instead of x this error will not occur.
principal square root function
Square root of 4 is both 2 and -2 according to the defination but then we cannot call this a function because it associates multiple outputs with a single input. We sometimes call it the principal square root function which is a function because it doesn't associate multiple values with a single input, principal square root of 4 is only 2 and not -2. Graphing principal square root function will give you a half parabola. You can play with graphs at desmos.com.
