Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \(\cosh x\) is an even function.

Step-by-Step Solution

Answer

\( \cosh x \) is an even function because \( \cosh(-x) = \cosh x \).

Step 1: Understanding Even Functions

A function is called an even function if it satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the function's domain. We need to show that this condition holds for the hyperbolic cosine function \( \cosh x \).

Step 2: Substitute Negative Argument

Consider \( \cosh(-x) \). By definition, the hyperbolic cosine function is \( \cosh x = \frac{e^x + e^{-x}}{2} \). Substitute \( -x \) into this expression: \( \cosh(-x) = \frac{e^{-x} + e^{x}}{2} \).

Step 3: Simplify the Expression

Notice that \( \frac{e^{-x} + e^{x}}{2} = \frac{e^{x} + e^{-x}}{2} \) because addition is commutative. Thus, \( \cosh(-x) = \cosh x \).

Step 4: Conclusion

Since \( \cosh(-x) = \cosh x \), by definition, \( \cosh x \) is an even function.

Key Concepts

Even Function

In mathematics, an even function is one that exhibits symmetry around the y-axis in its graph. This characteristic symmetry means that the function gives the same output for both a positive input and its negative counterpart. In formal terms, a function \( f(x) \) is considered even if \( f(x) = f(-x) \) for every \( x \) within the function’s domain. This property is crucial in various applications, especially when dealing with trigonometric and hyperbolic functions. The graph of an even function is always symmetrical with respect to the y-axis. This symmetry exists because the values on either side of the y-axis are identical.

Some common examples of even functions include \( x^2 \), \( \cos x \), and as you will see in this exercise, \( \cosh x \). Recognizing whether a function is even can greatly simplify solving equations, as it allows mathematicians to predict the behavior of the function at specific values without needing to graph every point.

cosh x

The hyperbolic cosine function \( \cosh x \) is a fundamental component of hyperbolic functions, which are analogous to certain trigonometric functions. The formula for \( \cosh x \) is defined as:

• \( \cosh x = \frac{e^x + e^{-x}}{2} \)

This function is similar to cosine function in trigonometry but instead of dealing with circle-based computations, hyperbolic functions are rooted in a hyperbola.

The symmetry and properties of \( \cosh x \) arise from the presence of both \( e^x \) and \( e^{-x} \), indicating that contributions from both exponential terms balance each other out. Consequently, \( \cosh x \) mirrors the behavior of the trigonometric cosine function, and just like its circular counterpart, it exhibits properties that allow it to be classified as an even function. Evaluating \( \cosh x \) at different values of \( x \) and \( -x \) shows that the two results are identical, further reinforcing its status as an even function.

Function Properties

Function properties help us understand the behavior and characteristics of functions. These properties include symmetry, periodicity, and specific rules for transformations. The hyperbolic cosine function \( \cosh x \) has several properties that make it reliable for mathematical modeling in various fields.

Some of the key properties of \( \cosh x \) include:

• **Evenness:** As demonstrated, \( \cosh x \) is an even function. It reflects symmetry about the y-axis, meaning \( \cosh x = \cosh(-x) \).
• **Non-negative values:** For all real numbers \( x \), \( \cosh x \geq 1 \). This property arises because both \( e^x \) and \( e^{-x} \) are always positive and contribute equally to the function's minimum value at zero.
• **Exponential growth:** For large absolute values of \( x \), \( \cosh x \) behaves exponentially, which is similar to the exponential function \( e^x \).
• **Derivative and Integral:** The derivative of \( \cosh x \) is \( \sinh x \), and its integral leads to other hyperbolic functions, showcasing its interconnection in calculus.

Understanding these properties is important because they form a basis for more complex mathematical problems. By leveraging these characteristic attributes, one can simplify computations and predict function behavior easily.