Question

Consider the functions f(x)={x^2}, g(x)={x}^2, and h(x)=x*{x} where {x}=x-[x] is the fractional part of x, also denoted by frac(x).

Answer 1

The student's question pertains to power functions and the properties of even and odd functions in mathematics. Specifically, the question includes examples of how to manipulate powers, the integral properties of odd functions, and the effects of translating functions along the x-axis.

Step-by-step explanation:

The question revolves around the concept of functions in mathematics, specifically concerning power functions and properties of even and odd functions.

An example of understanding a power function can be seen with 5¹ = 5, which implies that x² = √x, indicating that x squared is the same as the square root of x. This suggests how to deal with fractional powers and extracting roots from powers. When comparing odd and even functions, we see that the product of an even function with an even function remains even (e.g., x² * e^-x²), and similarly for odd functions. However, an odd function multiplied by an even function results in an odd function (e.g., x*e^-x²).

Moreover, an important property of odd functions is that their integral over the entire space equals zero due to the symmetrical nature of their graph with respect to the x-axis. This principle is used in various mathematical applications.

Finally, an aspect of translating functions is also discussed, illustrating how f(x - d) translates a function in the positive x-direction by d units, whereas f(x + d) translates it in the negative x-direction by the same amount.