Part 3/6:
Graphically, the function appears smooth and continuous, curving upwards as ( x ) increases, further demonstrating how power functions behave.
Values Between Zero and One
When examining values of ( x ) between zero and one, for instance at ( x = 0.5 ), we explore ( 3^{0.5} ), which calculates approximately to ( 1.73 ). This is a critical finding that illustrates how exponentiation works within fractional domains—where the output value remains greater than one when raised to fractional powers (remembering that ( a^a ) is tricky when ( a < 1 )).
Calculating points like ( 3^{1/4} ) yields even smaller results, approaching 1 but remaining greater than it, indicating the continuous trend toward unity.