Problem 56 - Thomas Calculus

Question

In Exercises \(47-58,\) say whether the function is even, odd, or neither. Give reasons for your answer. $$ h(t)=\left|t^{3}\right| $$

Step-by-Step Solution

Answer

The function \(h(t) = |t^3|\) is even because \(h(-t) = h(t)\).

Step 1: Understanding Even and Odd Functions

A function is even if for every input \( x \), \( f(-x) = f(x) \). A function is odd if for every input \( x \), \( f(-x) = -f(x) \). If neither of these conditions are met, the function is neither even nor odd.

Step 2: Substitute -t into the Function

To test if the function is even or odd, substitute \(-t\) for \(t\) in the given function. The function becomes \(h(-t) = \left| (-t)^3 \right|\).

Step 3: Simplify the Expression

Simplify \(h(-t) = \left| (-t)^3 \right|\). Note that \((-t)^3 = -(t^3)\), but since it is inside the absolute value function, \(\left| -(t^3) \right| = \left| t^3 \right|\), which is equivalent to \(h(t)\).

Step 4: Conclusion on Even Function

Since \(h(-t) = h(t)\), the function satisfies the condition for being an even function. Therefore, \(h(t)\) is even.

Key Concepts

Absolute Value Function

The absolute value function is a special mathematical concept that simplifies dealing with negative numbers. It is denoted by vertical bars, such as in \( |x| \). The absolute value of a number represents its distance from zero on the number line and it is always non-negative.

Key characteristics of the absolute value function include:

• If \( x \) is positive, \( |x| = x \).
• If \( x \) is zero, \( |x| = 0 \).
• If \( x \) is negative, \( |x| = -x \); in other words, take the negative of \( x \) to make it positive.

Absolute value functions are useful in ensuring outputs are always non-negative regardless of the sign of the input. In our example, we have \( h(t) = |t^3| \). Here, the absolute value ensures we are considering the positive magnitude of \( t^3 \), regardless of whether \( t \) is positive or negative. This characteristic plays a vital role in determining the symmetry of the function.

Function Properties

Functions have various properties that help us understand their behavior and characteristics. Important function properties include domain, range, and continuity, among others.

In this exercise, we are primarily concerned with the symmetry properties related to even and odd functions. To determine whether a function is even or odd, we analyze its algebraic behavior under specific transformations. Let's summarize these properties:

• An *even function* satisfies \( f(-x) = f(x) \) for every \( x \) in the domain. It is symmetric about the y-axis.
• An *odd function* satisfies \( f(-x) = -f(x) \) for every \( x \) in the domain. It is symmetric about the origin.

For the given function \( h(t) = |t^3| \), substituting \( -t \) into the function produces the same value as \( h(t) \). This illustrates a key property: the preserved output after transformation indicates symmetry consistent with an even function. This knowledge will guide further understanding of its symmetry.

Function Symmetry

Symmetry in functions is a fundamental aspect that reflects how a function’s output behaves when its inputs undergo specific transformations.

There are several types of symmetry we often consider in mathematics, specifically when classifying functions as even or odd. Let’s explore this a bit more:

• *Even Symmetry*: If \( f(x) \) is equal to \( f(-x) \) for every \( x \), the function is said to have even symmetry, implying it is symmetrical around the y-axis.
• *Odd Symmetry*: If \( f(x) = -f(-x) \), then the function has odd symmetry, meaning it is symmetric about the origin in a rotational sense.

When examining the function \( h(t) = |t^3| \), we notice that substituting negative inputs results in equivalent outputs to their positive counterparts, indicative of even symmetry. This is because the cube of a negative number retains the absolute value of the outcome of the cube of the positive number; hence, \( |(-t)^3| = |t^3| \). Therefore, the function is even, owing to its symmetry in reflecting y-axis properties.