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) \(x+\sin x\)

Short answer

(a) Odd, (b) Even, (c) Odd, (d) Even, (e) Even, (f) Odd.

Step-by-step solution

Understanding Even and Odd Functions

A function is **even** if for all values of the variable, \( f(-x) = f(x) \). This means its graph is symmetric about the y-axis. A function is **odd** if \( f(-x) = -f(x) \), which means its graph is symmetric about the origin. If neither condition is met, the function is neither even nor odd.

Analyze Function (a): \( t \sin t \)

Compute \((t \sin t)(-x) = -x \sin(-x) = -x(-\sin x) = x \sin x\). Compare to \((t \sin t)(x) = x \sin x\). Since \(f(-x) = -f(x)\), it is an odd function.

Analyze Function (b): \( \sin^2 t \)

Compute \((\sin^2 t)(-x) = (\sin(-x))^2 = (-\sin x)^2 = \sin^2 x\). Since \(f(-x) = f(x)\), it is an even function.

Analyze Function (c): \( \csc t \)

Compute \((\csc t)(-x) = \csc(-x) = \frac{1}{\sin(-x)} = -\frac{1}{\sin(x)}\). Since \(f(-x) = -f(x)\), it is an odd function.

Analyze Function (d): \( |\sin t| \)

Compute \((|\sin t|)(-x) = |\sin(-x)| = |-\sin x| = |\sin x|\). Since \(f(-x) = f(x)\), it is an even function.

Analyze Function (e): \( \sin(\cos t) \)

Compute \((\sin(\cos t))(-x) = \sin(\cos(-x)) = \sin(\cos x)\). Since \(f(-x) = f(x)\), it is an even function.

Analyze Function (f): \( x + \sin x \)

Compute \((x + \sin x)(-x) = -x + \sin(-x) = -x - \sin x\). Since \(f(-x) = -(f(x))\), it is an odd function.

Key concepts

Function Symmetry

Understanding the symmetry of functions is key to determining whether a function is even, odd, or neither. This involves comparing the function's behavior when negative inputs are used:

• Even functions exhibit y-axis symmetry. This means the graph of the function looks the same when flipped over the y-axis. Mathematically, for a function \( f(x) \) to be even, it must satisfy \( f(-x) = f(x) \) for all x in the function's domain.
• Odd functions display origin symmetry, meaning the graph can be rotated 180 degrees around the origin and appear unchanged. A function is odd if \( f(-x) = -f(x) \) holds true for all x.
• Some functions do not fit into these categories and are known as "neither" functions. They are asymmetric and do not meet the criteria for being even or odd.

It's useful to recognize these symmetries as they can simplify solving equations and understanding graphs.

Graphical Analysis

Graphical analysis plays a critical role when determining the symmetry of functions. By examining graphs, you can visually assess whether they are symmetric about the y-axis or the origin:

• Even function graphs are mirror images on either side of the y-axis. If you fold the graph along the y-axis, the halves will match perfectly.
• Odd function graphs will look identical even if you performed a 180-degree rotation about the origin. This reflects the operation \( f(-x) = -f(x) \) visually.
• Neither function graphs will not exhibit these symmetries, indicating they lack even or odd characteristics.

Graphical analysis helps in understanding and categorizing functions and provides a visual means of verification.

Mathematical Reasoning

Mathematical reasoning is necessary to confirm function symmetry analytically. This method involves substituting \(-x\) into a function and interpreting the results:

• For a function \( f(x) \) to be **even**, you must demonstrate that \( f(-x) = f(x) \). You can achieve this by substituting \(-x\) into every x and simplifying to see if it equals the original function.
• For a function to be **odd**, substituting \(-x\) must yield \(-f(x)\). If upon substitution, the operation results in negative of the original function, then the function is odd.
• If neither condition is satisfied, the function is inherently asymmetric and neither even nor odd.

This analytical approach ensures a precise identification of function symmetry and is essential when graphical inspection alone may not be sufficient.