Final answer:
The piecewise function for |x^5 - x^3| breaks down into two cases:
=> f(x) = x^5 - x^3 for x >= 0,
=> f(x) = -x^5 + x^3 for x < 0, based on the sign of x^5 - x^3.
Step-by-step explanation:
To express |x^5 - x^3| as a piecewise function, we first need to understand that the absolute value function outputs the same value for positive input but negates negative input, ensuring all outputs are non-negative.
Thus, the piecewise function for |x^5 - x^3| breaks down into two cases depending on the sign of x^5 - x^3.
For values of x that make x^5 - x^3 non-negative, we can directly write the function without absolute value bars:
f(x) = x^5 - x^3 for x^5 - x^3 ≥ 0
For values of x that make x^5 - x^3 negative, we negate the expression to make it non-negative:
f(x) = -(x^5 - x^3) for x^5 - x^3 < 0
We know that x^5 and x^3 are both odd functions, where odd powers maintain the sign of x. This means that the difference x^5 - x^3 will be non-negative when x is non-negative.
Conversely, for negative values of x, the difference becomes negative. Thus, the piecewise function simplifies to:
- f(x) = x^5 - x^3 for x ≥ 0
- f(x) = -(x^5 - x^3) = -x^5 + x^3 for x < 0