Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x/(x² − 1)

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To determine if a function is even, odd, or neither, we need to evaluate the function at -x and compare it to the original function g(x).

Calculate g(-x) by substituting -x into the function: g(-x) = (-x)/((-x)² - 1).

Simplify g(-x): Since (-x)² = x², we have g(-x) = -x/(x² - 1).

Compare g(-x) with g(x): g(x) = x/(x² - 1) and g(-x) = -x/(x² - 1). Notice that g(-x) = -g(x).

Since g(-x) = -g(x), the function g(x) is odd. A function is odd if g(-x) = -g(x) for all x in the domain of the function.

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주요 개념

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Even Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of an even function is symmetric with respect to the y-axis. A common example of an even function is f(x) = x², where substituting -x yields the same output as substituting x.

Odd Functions

A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of an odd function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, where substituting -x results in the negative of the output for x.

Function Analysis

Analyzing a function involves evaluating its behavior under transformations, such as substituting -x for x. This process helps determine whether the function is even, odd, or neither. For the function g(x) = x/(x² - 1), one must compute g(-x) and compare it to g(x) and -g(x) to classify its symmetry.

Even and Odd FunctionsIn Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.g(x) = x/(x² − 1)