Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following are odd functions? Even functions? Neither? (a) \(t \sin t\) (b) \(\sin ^{2} t\) (c) \(\csc t\) (d) \(|\sin t|\) (e) \(\sin (\cos t)\) (f) \(x+\sin x\)

Step-by-Step Solution

Answer

(a) Odd, (b) Even, (c) Odd, (d) Even, (e) Even, (f) Neither.

Step 1: Understand Even and Odd Functions

A function is even if for every point \(a\) in its domain, \(f(-a) = f(a)\), resulting in symmetry around the y-axis. A function is odd if for every point \(a\), \(f(-a) = -f(a)\), showing symmetry around the origin. A function that doesn't fit either of these symmetries is neither even nor odd.

Step 2: Examine (a): t sin t

Evaluate \(f(t) = t \sin t\). Compute \(f(-t) = (-t) \sin(-t) = t \sin t\), which shows that \(f(-t) = -f(t)\). Thus, \(t \sin t\) is an odd function.

Step 3: Examine (b): sin^2 t

Evaluate \(f(t) = \sin^2 t\). Compute \(f(-t) = \sin^2(-t) = \sin^2 t\), which shows that \(f(-t) = f(t)\). Thus, \(\sin^2 t\) is an even function.

Step 4: Examine (c): csc t

Evaluate \(f(t) = \csc t\). Compute \(f(-t) = \csc(-t) = -\csc t\), which shows that \(f(-t) = -f(t)\). Thus, \(\csc t\) is an odd function.

Step 5: Examine (d): |sin t|

Evaluate \(f(t) = |\sin t|\). Compute \(f(-t) = |\sin(-t)| = |-\sin t| = |\sin t|\), which shows that \(f(-t) = f(t)\). Thus, \(|\sin t|\) is an even function.

Step 6: Examine (e): sin(cos t)

Evaluate \(f(t) = \sin(\cos t)\). Compute \(f(-t) = \sin(\cos(-t)) = \sin(\cos t)\), which shows that \(f(-t) = f(t)\). Thus, \(\sin(\cos t)\) is an even function.

Step 7: Examine (f): x + sin x

Evaluate \(f(x) = x + \sin x\). Compute \(f(-x) = -x + \sin(-x) = -x - \sin x\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), \(x + \sin x\) is neither odd nor even.

Key Concepts

function symmetry

In mathematics, identifying whether a function is even, odd, or neither, largely revolves around understanding symmetry. Functions that exhibit symmetry in their graphs around the y-axis are termed as even. To determine if a function is even, you need to verify that for every input value \(a\), \(f(-a) = f(a)\).
Odd functions, on the other hand, have graphs that are symmetric about the origin. If for every \(a\), \(f(-a) = -f(a)\), the function qualifies as odd. Knowing how to recognize these symmetries helps in predicting the behavior of functions.

• **Even Function**: Symmetric around the y-axis, such as \((x) = (-x)\) for all \(x\).
• **Odd Function**: Symmetric about the origin, indicated by \((x) = -(-x)\).

Not every function will fall neatly into these categories; some may not exhibit any symmetry, and they are considered neither even nor odd.

mathematical symmetry

Mathematical symmetry is not limited to visual patterns; it is a broad concept that applies to various functions as well. Symmetry, in the mathematical sense, often implies a type of balance or equivalence. In the context of functions, symmetry pertains to how values of a function at points opposite one another behave.
Comprehending symmetry ensures a greater understanding of the predictability and continuity of functions. By analyzing symmetry, we find insights into a function's derivative behavior, integral properties, and more.

• **Reflection Symmetry (Even Functions)**: Reflected across the y-axis, making images mirror-like.
• **Rotational Symmetry (Odd Functions)**: Achieved when spinning a graph 180 degrees yields the same graph.

These properties don't just guide us algebraically but also aid in simplifying complex mathematical calculations by recognizing inherent patterns.

trigonometric functions

Trigonometric functions are a class of functions that exhibit interesting patterns of symmetry. These functions include sine, cosine, tangent, secant, cosecant, and cotangent, each with unique attributes of symmetry that aid in their calculation and graphical representation.
Understanding the symmetry of these functions simplifies solving equations and performing transformations. For instance, the sine function \(( heta) = ( heta + 2\pi)\) is periodic with a period of \(2\pi\), revealing its consistent nature across rotations.

• **Sine and Tangent**: Both behave as odd functions, showcasing origin symmetry.
• **Cosine**: This is an even function, reflecting its symmetry along the y-axis.

Grappling with these concepts makes it easier to delve into advanced topics like Fourier transformations and complex number calculations, where trigonometric functions play a central role.