Even and Odd Functions

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

h(t) = |t³|

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To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. A function f(t) is even if f(-t) = f(t) for all t in the domain, and it is odd if f(-t) = -f(t) for all t in the domain.

Consider the given function h(t) = |t³|. First, let's find h(-t) by substituting -t into the function: h(-t) = |-t³|.

Since the cube of a negative number is negative, we have (-t)³ = -(t³). Therefore, |-t³| = |-1 * t³| = |t³|, because the absolute value negates the negative sign.

Now, compare h(-t) with h(t): h(-t) = |t³| and h(t) = |t³|. Since h(-t) = h(t), the function h(t) is even.

Thus, the function h(t) = |t³| is an even function because it satisfies the condition h(-t) = h(t) for all t in its domain.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 the function is symmetric with respect to the y-axis. For example, the function f(x) = x² is even because f(-x) = (-x)² = x².

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 the function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, as f(-x) = (-x)³ = -x³.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is neither even nor odd because it does not satisfy the conditions for either classification. For instance, h(t) = |t³| results in h(-t) = |-t³| = |t³|, which is equal to h(t), indicating it is even, but the cubic term's sign negation complicates its classification.

Even and Odd FunctionsIn Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.h(t) = |t³|