Problem 66 - Algebra and Trigonometry

Question

Compositions of Odd and Even Functions Suppose that $$h=f \circ g$$ If \(g\) is an even function, is \(h\) necessarily even? If \(g\) is odd and \(f\) \(h\) odd? What if \(g\) is odd and \(f\) is odd? What if \(g\) is odd and \(f\) is even?

Step-by-Step Solution

Answer

When both \(f\) and \(g\) are odd, \(h\) is odd; when \(g\) is odd and \(f\) is even, \(h\) is even. Parity depends on \(f\) when \(g\) is even.

Step 1: Understanding Even and Odd Functions

An even function satisfies the condition \(f(x) = f(-x)\) for all \(x\). An odd function satisfies \(f(x) = -f(-x)\) for all \(x\). Function \(h(x) = (f \circ g)(x)\) means \(h(x) = f(g(x))\).

Step 2: Case 1 - g is Even

If \(g\) is even, then \(g(x) = g(-x)\). For \(h\) to be even, \(h(x) = h(-x)\) must be true, which implies \(f(g(x)) = f(g(-x))\). Since \(g(x) = g(-x)\), this is true regardless of whether \(f\) is odd or even. Therefore, \(h\) could be even depending on \(f\).

Step 3: Case 2 - g and f both Odd

If both \(g\) and \(f\) are odd, then \(g(x) = -g(-x)\) and \(f(x) = -f(-x)\). For \(h(x) = f(g(x))\), we have \(h(-x) = f(g(-x)) = f(-g(x)) = -f(g(x)) = -h(x)\). Thus, \(h\) is also odd.

Step 4: Case 3 - g Odd and f Even

If \(g\) is odd and \(f\) is even, then \(g(x) = -g(-x)\) and \(f(x) = f(-x)\). For \(h(x) = f(g(x))\), we have \(h(-x) = f(g(-x)) = f(-g(x)) = f(g(x)) = h(x)\). Thus, \(h\) is even.

Step 5: Case 4 - g Odd and f Undetermined

If \(g\) is odd and \(f\)'s parity is not specified, \(g(x) = -g(-x)\). The parity of \(h(x) = f(g(x))\) depends entirely on \(f\). Given the other cases, without more information about \(f\), we cannot determine \(h\)'s parity. Therefore, without specification, \(h\)'s parity remains undetermined.

Key Concepts

Odd Functions

An odd function is a type of function with a unique symmetrical property. In precise terms, a function, say \( f(x) \), is considered odd if it satisfies the condition \( f(-x) = -f(x) \) for every \( x \) in its domain.
This means that the graph of an odd function is symmetric with respect to the origin.
Odd functions flip over the origin, illustrating this peculiar symmetry.When you think about functions like sine, which is an odd function, for every point on the graph, the point reflected across both the x-axis and the y-axis (origin) is also on the graph.
This origin symmetry leads to interesting properties, such as:

• The integral of an odd function over a symmetric interval around zero is zero.
• If you rotate the graph 180 degrees around the origin, it will look the same.

In the context of composite functions, if both functions involved are odd, the resulting function is also odd, maintaining this symmetry.

Even Functions

Unlike odd functions, even functions exhibit symmetry about the y-axis. A function \( f(x) \) is considered even when \( f(x) = f(-x) \) for all \( x \) within its domain.
This means if you graph an even function, a mirror placed along the y-axis would let you see that both sides are identical reflections of each other.A common example of an even function is the cosine function or any function involving only even powers of \( x \), like \( x^2 \).
Some interesting properties of even functions include:

• The graph remains unchanged when reflected over the y-axis.
• Often, the Fourier series of even functions include only cosine terms.

For composite functions, if the internal function \( g(x) \) is even and does not change the symmetry of \( f(x) \) significantly, \( h(x) = f(g(x)) \) could also be even, especially when \( f(x) \) itself is even or does not have odd properties that disrupt the symmetry.

Function Parity

Function parity refers to the inherent symmetry properties of functions, falling under two main categories: even and odd.
While some functions are neither distinctly even nor odd, understanding parity is crucial for analyzing composite functions and their behavior.With composite functions like \( h(x) = f(g(x)) \), function parity becomes an essential tool:

• If \( g(x) \) is even, the parity of \( h(x) \) depends heavily on \( f(x) \). Whether even or odd, it can define whether \( h(x) \) retains symmetry.
• If both \( g(x) \) and \( f(x) \) are odd, \( h(x) \) will likely be odd, due to the nature of composition preserving the odd trait.
• If \( g(x) \) is odd and \( f(x) \) is even, \( h(x) \) tends to be even, aligning with the y-axis symmetry contributed by \( f(x) \).
• If the parity of \( f(x) \) is undefined, \( h(x) \)'s parity cannot be simply deduced.

The study of parity helps break down complex problems into simpler, symmetrical parts, aiding mathematical analysis and problem-solving.