Final answer:
The compositions f(g(x)) and g(f(x)) simplify to x, showing that g(x) acts as an inverse function to f(x). An even function remains even when multiplied by another even function. Conversely, the product of an odd and even function is odd.
Step-by-step explanation:
Composition of Functions and Even/Odd Functions
To solve the composition of functions, f(g(x)), you substitute g(x) into f(x). Here, substituting g(x) = x - 2 into f(a) where a is now x - 2, we have f(g(x)) = (x - 2) + 2 which simplifies to x. Similarly, for g(f(x)), f(x) is substituted into g(x), which results in g(f(x)) = (x + 2) - 2, also simplifying to x. Hence, it is shown that g(x) acts as an inverse function to f(x).
Regarding even and odd functions, an even function is symmetric about the y-axis, and the product of two even functions, or two odd functions, is again an even function. An example of an even function is x² multiplied by e-x². An odd function is symmetric with respect to the origin, and the product of an odd and an even function is an odd function, such as x multiplied by e-x². Moreover, the integral of an odd function over all space is zero due to the symmetry of its graph above and below the x-axis.