Analyzing the Nature of the Displacement Function g(t)

    Analyzing the Nature of the Displacement Function g(t) Thesis Statement In analyzing the given displacement function, g(t) = \frac{10t^3}{12t^2+53}, it is crucial to understand the concepts of even and odd functions mathematically and graphically to determine the nature of symmetry displayed by the function. Even Function Analysis An even function is defined as f(x) = f(-x) for all x in the domain. To determine if the given function g(t) is even, Alex can substitute -t for t in the function and simplify to check if it equals the original function. If g(t) = g(-t), then the function displays even symmetry. Graphical Analysis of Symmetry - Even Symmetry: A function has even symmetry if it is symmetric with respect to the y-axis. For g(t) to possess even symmetry, its graph should be symmetric with respect to the y-axis. This means that replacing t with -t in the function should result in the same y-value. - Odd Symmetry: A function has odd symmetry if f(x) = -f(-x) for all x in the domain. In graphical terms, a function with odd symmetry is symmetric about the origin (0,0). When t is replaced by -t in the function g(t), if f(-t) = -f(t), then it displays odd symmetry. - Neither: If the function does not satisfy the criteria for even or odd symmetry, it possesses neither symmetry. Conclusion By understanding the mathematical definitions of even and odd functions and interpreting them graphically, Alex can determine whether the displacement function g(t) = \frac{10t^3}{12t^2+53} displays even symmetry, odd symmetry, or neither. This analysis will provide insights into the nature and behavior of the function, aiding in Alex's examination of the mechanical system.

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