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|

Verified step by step guidance

1. 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.
2. Consider the given function h(t) = |t³|. First, let's find h(-t) by substituting -t into the function: h(-t) = |-t³|.
3. 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.
4. 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.
5. 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.

Key Concepts

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.