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?

b. f/g

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First, recall the definitions of even and odd functions. An even function satisfies f(x) = f(-x) for all x, while an odd function satisfies g(x) = -g(-x) for all x.

Consider the function h(x) = f(x)/g(x). To determine if h(x) is even or odd, we need to evaluate h(-x) and compare it to h(x).

Calculate h(-x): h(-x) = f(-x)/g(-x). Since f is even, f(-x) = f(x). Since g is odd, g(-x) = -g(x). Therefore, h(-x) = f(x)/(-g(x)).

Simplify h(-x): h(-x) = -f(x)/g(x). Notice that h(-x) = -h(x), which is the definition of an odd function.

Conclude that the function h(x) = f(x)/g(x) is odd, given that f is even and g is odd.

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Even and Odd Functions

An even function satisfies 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 Division

When dividing two functions, such as f/g, the resulting function is defined only where g(x) is not zero. The properties of the original functions (even or odd) can influence the nature of the quotient, but one must also consider the points where the denominator may affect the overall function's parity.

Parity of Combined Functions

The parity of combined functions can be determined by analyzing their definitions. For instance, the quotient f/g will be even if both f and g are even, or if f is even and g is odd. Conversely, it will be odd if f is odd and g is even. This requires careful consideration of the definitions and properties of the functions involved.

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?b. f/g