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: Understanding Even and Odd Functions

An even function satisfies the condition \(f(-x) = f(x)\), meaning it is symmetric about the y-axis. An odd function satisfies the condition \(f(-x) = -f(x)\), meaning it is symmetric about the origin. Any function that does not meet either of these criteria is neither even nor odd.

Step 2: Analyze \(\cot t + \sin t\)

For \(f(t) = \cot t + \sin t\), compute \(f(-t) = \cot(-t) + \sin(-t) = -\cot t - \sin t = -(\cot t + \sin t)\). Since \(f(-t) = -f(t)\), this function is odd.

Step 3: Analyze \(\sin^3 t\)

For \(f(t) = \sin^3 t\), compute \(f(-t) = \sin^3(-t) = (-\sin t)^3 = -\sin^3 t = -f(t)\). Since \(f(-t) = -f(t)\), this function is odd.

Step 4: Analyze \(\sec t\)

For \(f(t) = \sec t\), compute \(f(-t) = \sec(-t) = \sec t\). Since \(f(-t) = f(t)\), this function is even.

Step 5: Analyze \(\sqrt{\sin^4 t}\)

For \(f(t) = \sqrt{\sin^4 t}\), \(f(t) = |\sin^2 t|\) because the square and square root eliminate the sign. Compute \(f(-t) = |\sin^2(-t)| = |\sin^2 t| = f(t)\). Therefore, this function is even.

Step 6: Analyze \(\cos(\sin t)\)

For \(f(t) = \cos(\sin t)\), compute \(f(-t) = \cos(\sin(-t)) = \cos(-\sin t) = \cos(\sin t) = f(t)\). Therefore, this function is even.

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

For \(f(x) = x^2 + \sin x\), compute \(f(-x) = (-x)^2 + \sin(-x) = x^2 - \sin x\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), this function is neither even nor odd.

Key Concepts

Symmetry of Functions

Functions can have various types of symmetry that tell us how they behave when we reflect them across certain axes. Two common types of symmetry are:

• **Symmetry about the y-axis**: If a function is symmetric about this axis, it's called an **even function**. It satisfies the condition: \(f(-x) = f(x)\). This means that if you fold the graph of the function along the y-axis, both sides will match perfectly.
• **Symmetry about the origin**: If a function is symmetric this way, it's called an **odd function**. It fulfills the condition: \(f(-x) = -f(x)\). This reflects the function through both axes, showing a kind of mirror image about the origin.

Understanding these types of symmetry is crucial when analyzing and solving problems involving even and odd functions. Some functions may not possess either type of symmetry, meaning they are neither even nor odd.

Trigonometric Functions

Trigonometric functions are a special class of functions that relate angles to ratios of sides in a right-angled triangle. Important trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), among others.
Each of these functions has unique properties related to symmetry:

• **Sine**: This trigonometric function is odd because \(\sin(-x) = -\sin(x)\).
• **Cosine**: This function is even, adhering to \(\cos(-x) = \cos(x)\).
• **Tangent**: An odd function because \(\tan(-x) = -\tan(x)\).

Knowledge of these properties can simplify the process of determining the symmetry of more complicated composite functions that include trigonometric components.

Function Analysis

Analyzing a function involves examining its properties such as symmetry, zeros, and period (if it's periodic). In this context:

• **Breaking down composite functions**: Evaluate how each part of the function behaves, especially if it involves additions or multiplications of known functions. For example, \(f(t) = \cot t + \sin t\) can be examined by looking at the behavior of \(\cot t\) and \(\sin t\) separately.
• **Studying power functions**: When analyzing functions like \(\sin^3 t\), it's critical to understand how exponents affect symmetry. Thus, \(\sin^3 t\) remains odd due to the properties of sine and the exponent.
• **Applying transformations**: Consider transformations like the square or square root that can change function parity. For example, while \(\sin t\) is odd, \(\sin^4 t\) becomes even due to squaring.

Function analysis often involves both algebraic manipulation and a grasp of function properties.

Function Parity Check

Function parity checks involve determining if a function is even, odd, or neither. This is done by assessing the conditions:

• **Check for Evenness**: Evaluate \(f(-x)\) and see if it equals \(f(x)\). If so, the function is even.
• **Check for Oddness**: Calculate \(f(-x)\) and verify whether it equals \(-f(x)\). If true, the function is odd.
• **If Neither**: If neither condition holds, then the function is neither even nor odd.

For example, with the function \(x^2 + \sin x\), upon evaluating \(f(-x) = x^2 - \sin x\), we find that it doesn't match \(f(x)\) or \(-f(x)\), making it neither even nor odd. This check is a systematic way to categorize functions based on their symmetry properties.