Problem 89 - Precalculus

Question

In calculus the following two functions are studied: $$ \sinh x=\frac{e^{x}-e^{-x}}{2} \quad \text { and } \quad \cosh x=\frac{e^{x}+e^{-x}}{2} $$ Determine whether \(f(x)=\sinh x\) is an even function or an odd function.

Step-by-Step Solution

Answer

\( f(x) = \sinh x \) is an odd function.

Step 1: Understanding Even and Odd Functions

A function is called even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \), meaning it is symmetric about the y-axis. A function is called odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \), which implies it has rotational symmetry about the origin. We will apply these definitions to \( \sinh x \).

Step 2: Substitute \(-x\) into \(\sinh x\)

To determine if \( f(x) = \sinh x \) is odd or even, we substitute \(-x\) for \(x\): \[ \sinh(-x) = \frac{e^{-x} - e^{x}}{2}. \]

Step 3: Simplify the Expression

Start simplifying the expression for \( \sinh(-x) \): \[ \sinh(-x) = \frac{1}{2} (e^{-x} - e^{x}) = \frac{-1}{2} (e^{x} - e^{-x}). \] Notice that the expression \( \frac{-(e^x - e^{-x})}{2} \) is equal to \(-\sinh(x)\).

Step 4: Compare with \(\sinh x\)

We have that \( \sinh(-x) = -\frac{e^{x} - e^{-x}}{2} = -\sinh(x) \). This step confirms that \( \sinh(-x) = -\sinh(x) \).

Step 5: Conclusion

Since \( \sinh(-x) = -\sinh(x) \), \( \sinh x \) satisfies the definition of an odd function. Therefore, \( f(x) = \sinh x \) is an odd function.

Key Concepts

Even and Odd Functions

In mathematics, functions are often categorized into even and odd functions based on their symmetry properties about the origin or the y-axis. Understanding these classifications helps simplify function analysis and identify key characteristics.

• **Even Functions**: A function is considered even if, when you replace \( x \) with \( -x \), the function remains unchanged. This means that \( f(-x) = f(x) \). Such functions are symmetric about the y-axis. Examples include \( f(x) = x^2 \) and \( ext{cos } x \).
• **Odd Functions**: A function is classified as odd if replacing \( x \) with \( -x \) results in the negation of the original function, so \( f(-x) = -f(x) \). These functions exhibit rotational symmetry about the origin. Examples include \( f(x) = x^3 \) and \( ext{sin } x \).

Recognizing these properties allows mathematicians and students to predict function behavior without extensive calculations, streamlining problem-solving processes.

Sinh Function

The \( ext{sinh} \) function, also known as the hyperbolic sine, has a special place in calculus due to its unique properties and applications in various mathematical and physical problems.

• The hyperbolic sine function \( ext{sinh}(x) \) is defined mathematically as:\[ \text{sinh}(x) = \frac{e^x - e^{-x}}{2}. \]
• This function relates to the exponential function, making it valuable for analyzing growth and decay processes in physics.
• By substituting \(-x\) into the formula, we calculate \( ext{sinh}(-x) = \frac{e^{-x} - e^x}{2} \), which simplifies to \(-\text{sinh}(x)\). This shows a key characteristic of the \( ext{sinh} \) function: it is odd.

The hyperbolic sine is particularly interesting because, unlike the regular sine and cosine, it derives from the exponential function, providing a bridge between trigonometry and calculus. Students studying hyperbolic functions often find these insights invaluable for understanding larger topics in advanced mathematics.

Symmetry in Functions

Symmetry in functions is a fundamental concept in mathematics that helps in understanding and analyzing mathematical models. Symmetry can reveal insights into the nature of functions and their graphs.

• A function that maintains identical values when mirrored across a particular line (like the y-axis) shows even symmetry. Such functions have balanced and predictable shapes.
• Functions portraying symmetry about the origin are considered odd. These functions have rotational symmetry, where the function's curve rotates 180 degrees about the origin.
• For \( ext{sinh}(x) \), the calculated symmetry \( ext{sinh}(-x) = -\text{sinh}(x) \) confirms that it is an odd function, as it retains its properties upon a 180-degree rotation around the source.

By incorporating these symmetry properties, especially in hyperbolic functions like \( ext{sinh} \), mathematicians can simplify calculations and predict behavior over specific intervals, which is crucial for diverse applications in engineering and physics.