Part 5/6:
The function ( f(x) = x^{-1} ) simplifies to ( f(x) = 1/x ). The graph exhibits hyperbolic behavior, approaching zero as ( x ) tends toward infinity and diverging as ( x ) approaches zero.
For ( x < 0 ), the function reflects similar behavior but in the negative quadrant, reinforcing that it approaches zero from the negative side.
Conclusion
All the observed patterns in graphing power functions underscore their mathematical beauty and complexity. Power functions not only exhibit clear patterns based on the nature of their exponents, but they also play an integral role in calculus and higher mathematics. Understanding these concepts equips learners with valuable tools for further exploration of mathematical theories.