Problem 76 - Algebra and Trigonometry

Question

Determine whether the function is even, odd, or neither. \(f(x)=\sin x+\cos x\)

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand the Definitions

Recall that a function is even if \(f(-x) = f(x)\) for all \(x\) in the domain of \(f\). It is odd if \(f(-x) = -f(x)\). If neither of these conditions hold, the function is neither even nor odd.

Step 2: Find f(-x)

Calculate \(f(-x)\) by substituting \(-x\) into the function: \[f(-x) = \sin(-x) + \cos(-x)\]Using the identities \(\sin(-x) = -\sin x\) and \(\cos(-x) = \cos x\), we get:\[f(-x) = -\sin x + \cos x\]

Step 3: Compare f(-x) to f(x)

Compare the expressions for \(f(x)\) and \(f(-x)\):\[f(x) = \sin x + \cos x\]\[f(-x) = -\sin x + \cos x\]Notice that these expressions are not equal, so the function is not even.

Step 4: Check if Function is Odd

To check if the function is odd, see if \(f(-x) = -f(x)\):\[-f(x) = - (\sin x + \cos x) = -\sin x - \cos x\]Compare this with \(f(-x) = -\sin x + \cos x\). Since they are not the same, the function is not odd either.

Step 5: Conclusion

Since the function \(f(x) = \sin x + \cos x\) does not satisfy the conditions for being even or odd, it is neither even nor odd.

Key Concepts

Even Functions

Even functions have a unique mathematical symmetry. In essence, a function is classified as even when, for every input \(x\), the output remains the same if you input the opposite \(-x\). This is expressed as \(f(-x) = f(x)\) for all \(x\) in its domain. An intuitive example of this is the cosine function, \(f(x) = \cos x\), which showcases symmetry about the vertical \(y\)-axis. Graphically, this means if you fold the graph along the \(y\)-axis, the left and right sides will mirror each other perfectly.

• Even functions are symmetric about the \(y\)-axis.
• Common examples include \(y = x^2\) and \(y = \cos x\).

Understanding this concept is crucial because it simplifies analysis, equations, and understanding a function’s behavior over its domain.

Odd Functions

Odd functions stand apart with a different type of symmetry. For odd functions, the key property is that \(f(-x) = -f(x)\). This means for every \(x\), the function’s output when input is flipped to the negative, is just the negative of the original function's output. The sine function \(f(x) = \sin x\) is a classic example. It showcases symmetry around the origin, meaning if you rotate the graph 180 degrees around the origin, it looks the same.

• Odd functions are symmetric about the origin.
• Examples include \(y = x^3\) and \(y = \sin x\).

This concept helps identify how functions behave under transformations and is useful in solving trigonometric equations.

Function Symmetry

The concept of symmetry in functions helps simplify and understand many issues in mathematics. It relates to how functions behave when inputs are negated. Symmetry can be visualized graphically and analyzed algebraically with characteristics of even and odd functions. While even functions are symmetric with respect to the \(y\)-axis, odd functions exhibit symmetry about the origin.
Specifically:

• Even Function: \(y\)-axis symmetrical, \(f(-x) = f(x)\).
• Odd Function: Origin symmetrical, \(f(-x) = -f(x)\).
• Neither: Functions that don't satisfy these conditions have no specific symmetry.

This knowledge of symmetry aids in exploring function characteristics such as roots, intercepts, and transformations.

Sine Function

The sine function is a fundamental concept in trigonometry. Represented as \(f(x) = \sin x\), it is an odd function. This implies its deep link with origin symmetry, as \(\sin(-x) = -\sin x\). The sine function undergoes periodic oscillation, returning to the same value at intervals of \(2\pi\). Its graph forms a smooth wave that crosses the origin and repeats predictably.
Key features include:

• Period: \(2\pi\)
• Amplitude: Maximum height of \(1\)
• Symmetric about the origin

Understanding the sine function aides in tackling a range of trigonometric problems, including waves, vibrations, and circular motion studies.

Cosine Function

In the world of trigonometric functions, the cosine function, \(f(x) = \cos x\), is an essential player. Unlike the sine function, the cosine function is even, making it symmetric about the \(y\)-axis. This is defined by \(\cos(-x) = \cos x\). Cosine functions repeat their values every \(2\pi\), akin to the sine function, but are phase-shifted by \(\pi/2\) relative to it.
Important aspects include:

• Period: \(2\pi\)
• Amplitude: 1
• Centered around \(y\)-axis symmetry

This knowledge becomes a powerful tool when analyzing sound waves, electromagnetics, and alternating currents.