Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?


a. ƒg
b. ƒ³
c. ƒ(sin x)
d. g(sec x)
e. |g|

Verified step by step guidance

1. Recall that a function h(x) is odd if h(-x) = -h(x) for all x in its domain, and it is even if h(-x) = h(x).
2. For part (a), consider the product of two odd functions, ƒ and g. The product of two odd functions is even because (ƒg)(-x) = ƒ(-x)g(-x) = (-ƒ(x))(-g(x)) = ƒ(x)g(x).
3. For part (b), consider the cube of an odd function, ƒ³. Since odd functions raised to an odd power remain odd, ƒ³ is odd because (ƒ³)(-x) = (ƒ(-x))³ = (-ƒ(x))³ = -ƒ³(x).
4. For part (c), consider the composition of an odd function ƒ with the sine function, ƒ(sin x). Since sin(x) is an odd function, the composition of two odd functions is even, so ƒ(sin x) is even.
5. For part (d), consider the composition of an odd function g with the secant function, g(sec x). Since sec(x) is an even function, the composition of an odd function with an even function is odd, so g(sec x) is odd. For part (e), consider the absolute value of an odd function, |g|. The absolute value of any function is even because |g(-x)| = |g(x)|.

Key Concepts

Odd Functions

An odd function is defined by the property that f(-x) = -f(x) for all x in its domain. This means that the graph of an odd function is symmetric with respect to the origin. Understanding this property is crucial for determining the parity (even or odd) of combinations of odd functions, such as products or compositions.

Even Functions

An even function satisfies the condition f(-x) = f(x) for all x in its domain, indicating that its graph is symmetric about the y-axis. Recognizing how even and odd functions interact is essential for analyzing the parity of expressions involving these functions, especially when combined or transformed.

Function Composition and Products

The composition and product of functions can yield new functions whose parity can be determined by the parities of the original functions. For instance, the product of two odd functions is even, while the composition of an odd function with an even function retains the odd property. This concept is vital for evaluating the parity of the given expressions involving functions f and g.