Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

g. g ∘ f

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Understand the definitions: An even function f satisfies f(x) = f(-x) for all x in its domain, while an odd function g satisfies g(x) = -g(-x) for all x in its domain.

Consider the composition of functions g ∘ f, which means g(f(x)). We need to determine if this composition is even or odd.

To check if g ∘ f is even, verify if g(f(x)) = g(f(-x)) for all x. Since f is even, f(-x) = f(x), so g(f(-x)) = g(f(x)).

To check if g ∘ f is odd, verify if g(f(x)) = -g(f(-x)) for all x. Since f is even, f(-x) = f(x), so g(f(-x)) = g(f(x)). For g to be odd, g(f(x)) should equal -g(f(x)), which is only true if g(f(x)) = 0.

Conclude that g ∘ f is an even function because g(f(x)) = g(f(-x)) holds true for all x, given that f is even and g is applied to the same value f(x).

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Even and Odd Functions

An even function is defined by the property f(x) = f(-x) for all x in its domain, meaning its graph is symmetric about the y-axis. An odd function satisfies g(x) = -g(-x), indicating that its graph is symmetric about the origin. Understanding these definitions is crucial for analyzing the behavior of combined functions.

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For example, if we have functions f and g, the composition g ∘ f means we apply f first and then g to the result. This concept is essential for determining the properties of the resulting function when combining even and odd functions.

Properties of Composed Functions

When composing functions, the resulting function's parity (even or odd) can be determined by the parities of the original functions. Specifically, the composition of an even function with any function retains evenness, while the composition of an odd function with an even function results in an odd function. This understanding is key to solving the problem regarding the parity of g ∘ f.

Combining FunctionsAssume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?g. g ∘ f