Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following are odd functions? Even functions? Neither? (a) \(\cot t+\sin t\) (b) \(\sin ^{3} t\) (c) \(\sec t\) (d) \(\sqrt{\sin ^{4} t}\) (e) \(\cos (\sin t)\) (f) \(x^{2}+\sin x\)

Step-by-Step Solution

Answer

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

Step 1: Understand Odd and Even Functions

A function is odd if for every input \( x \), \( f(-x) = -f(x) \). It is even if \( f(-x) = f(x) \). Neither if it satisfies neither condition.

Step 2: Analyze Function (a) \( f(t) = \cot t + \sin t \)

Calculate \( f(-t) = \cot(-t) + \sin(-t) = -\cot t - \sin t = -(\cot t + \sin t) = -f(t) \). This shows it is an odd function.

Step 3: Analyze Function (b) \( f(t) = \sin^3 t \)

Calculate \( f(-t) = \sin^3(-t) = (-\sin t)^3 = -\sin^3 t = -f(t) \). Thus, it is an odd function.

Step 4: Analyze Function (c) \( f(t) = \sec t \)

Calculate \( f(-t) = \sec(-t) = \frac{1}{\cos(-t)} = \frac{1}{\cos t} = \sec t = f(t) \). Therefore, it is an even function.

Step 5: Analyze Function (d) \( f(t) = \sqrt{\sin^4 t} \)

Simplify first: \( \sqrt{\sin^4 t} = |\sin^2 t| = \sin^2 t\). Calculate \( f(-t) = \sin^2(-t) = (\sin^2 t) = \sin^2 t = f(t) \). It is even.

Step 6: Analyze Function (e) \( f(t) = \cos(\sin t) \)

Calculate \( f(-t) = \cos(\sin(-t)) = \cos(-\sin t) = \cos(\sin t) = f(t) \). Consequently, it is even.

Step 7: Analyze Function (f) \( f(x) = x^2 + \sin x \)

Calculate \( f(-x) = (-x)^2 + \sin(-x) = x^2 - \sin x \). Since \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), it is neither odd nor even.

Key Concepts

Odd Functions

Odd functions are intriguing because of their unique symmetry properties about the origin. Simply put, a function is considered odd if it satisfies this specific condition: \( f(-x) = -f(x) \). This means that if you plug in the negative of any input value \( x \), the output is also the negative of the original function value. Imagine flipping both the input and output in opposite directions.Here are some important traits of odd functions:

• They show rotational symmetry around the origin in their graphs.
• An easy example is the function \( f(x) = x^3 \). If you rotate the graph 180 degrees about the origin, it looks unchanged.
• In trigonometry, many functions are odd, like \( \sin x \) and \( \tan x \).

In our exercise, both \( \cot t + \sin t \) and \( \sin^3 t \) are odd functions. When you substitute \(-t\) into each and simplify, you see the functions satisfy \( f(-t) = -f(t) \). This confirms their odd nature.

Even Functions

Even functions shine with their reflectional symmetry. A function is deemed even if it holds true that \( f(-x) = f(x) \) for all applicable values of \( x \). This feature reveals that the function's graph remains unchanged when flipped over the y-axis.Core characteristics of even functions include:

• Their graphs are mirror images across the y-axis. If you fold the graph along the y-axis, the two sides will match perfectly.
• Common examples are functions like \( f(x) = x^2 \) or \( \cos x \), both reflecting symmetry about the vertical axis.
• Trigonometric even functions include \( \cos x \) and \( \sec x \).

Looking at the exercise, \( \sec t \), \( \sqrt{\sin^4 t} \), and \( \cos(\sin t) \) all meet the criteria of even functions. When we substitute \(-t\) and simplify, the expressions stay equivalent to their original form.

Neither Odd Nor Even

Sometimes, a function doesn't fit neatly into the categories of odd or even. When a function exhibits neither odd nor even properties, it means neither symmetry condition (for odd or even functions) holds true. This occurs when substituting \(-x\) gives results that are neither equal to \( f(x) \) nor equal to \(-f(x) \).Attributes of such functions include:

• Graphs that lack symmetry about both the origin and the y-axis.
• The presence of both even and odd terms can often lead a function to be neither. For instance, \( f(x) = x^2 + \sin x \) has a mix of an even term \( x^2 \) and an odd term \( \sin x \).
• Some polynomial functions may fall into this category if neither stated symmetry conditions apply.

In our exercise, the function \( x^2 + \sin x \) does not satisfy either symmetry condition. Here, substituting \(-x\) doesn’t give the same value nor does it yield the negation of the function, establishing it as neither odd nor even.