To understand the implications of g being an even function and h being an even function of g, let us first define what we mean by an even function.
An even function is one that satisfies the condition f(-x) = f(x) for all x in its domain. This symmetry about the y-axis means that the function produces the same output for both the positive and negative values of x.
Now, if we define g as an even function, we can write:
g(-x) = g(x)
Next, when we talk about h being an even function of gh is also symmetric around the y-axis. Therefore, for h applied to g, the following holds true:
h(g(-x)) = h(g(x))
Since we know that g(-x) = g(x), we can substitute this into the equation:
h(g(-x)) = h(g(x))- h(g(-x)) = h(g(x))
This means that when h is an even function of g, the property of evenness is preserved. Therefore, if you take any even function g and apply an even function h, the resulting function will also be even.
In conclusion, if both g and h are even functions, the composition or the resultant function of h applied to g remains even:
h(g(x)) is even
This relationship illustrates the concept of symmetry in mathematics and showcases the beauty of function properties when combined in specific ways.