Problem 67 - Algebra for College Students

Question

$$ \text { Solve the equation } y=\frac{e^{x}-e^{-x}}{2} \text { for } x \text { in terms of } y \text {. } $$

Step-by-Step Solution

Answer

\( x = \ln(y + \sqrt{y^2 + 1}) \).

Step 1: Understand the Hyperbolic Function

Recognize that the given equation can be rewritten in terms of hyperbolic functions. The function \( y = \frac{e^x - e^{-x}}{2} \) is the definition of the hyperbolic sine function, \( y = \sinh(x) \).

Step 2: Inverse Hyperbolic Sine Function

To solve for \( x \), use the inverse hyperbolic sine function. The inverse of \( y = \sinh(x) \) is \( x = \text{arsinh}(y) \) or \( x = \sinh^{-1}(y) \).

Step 3: Express in Terms of Logarithms

Recall the formula for the inverse hyperbolic sine, \( x = \sinh^{-1}(y) = \ln(y + \sqrt{y^2 + 1}) \). Use this formula to solve for \( x \).

Step 4: Write the Final Expression

Thus, the solution for \( x \) in terms of \( y \) is \( x = \ln(y + \sqrt{y^2 + 1}) \).

Key Concepts

Hyperbolic Sine Function

The hyperbolic sine function, often denoted as \( \sinh(x) \), is a mathematical function that shares a similar name with the more familiar sine function from trigonometry. However, unlike the trigonometric sine function, \( \sinh(x) \) relates to exponential functions. It's defined as: \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] This function can be visualized on the Cartesian plane, where it exhibits a smooth curve that extends infinitely in both directions, unlike the periodic nature of the trigonometric sine function.

• \( e^x \) and \( e^{-x} \) are the exponential functions involved in its definition.
• The function describes certain types of growth and decay and appears in various engineering and physics equations.

Often, it might be necessary to work backwards from \( \sinh(x) \) to find \( x \). That's when the inverse hyperbolic functions come in handy.

Solving Equations

Solving equations involving hyperbolic functions is akin to solving for unknowns in any mathematical equation. Suppose you are given the equation \( y = \sinh(x) \) and need to solve for \( x \). Here, you need to use the concept of inverse functions.

• Firstly, you would identify the given hyperbolic function and recognize it as \( y = \sinh(x) \).
• The goal is to isolate \( x \). Here, we use the inverse function method, where you apply the inverse hyperbolic sine function \( \text{arsinh}(y) \) or \( \sinh^{-1}(y) \).

Finding the inverse allows for the resolution of \( x \) in terms of \( y \), providing a clear path to the solution. In this particular problem, once the inverse function is applied, \( x \) can be expressed through simplifying the equation.

Logarithmic Functions

Logarithmic functions come into play when dealing with inverse hyperbolic functions like \( \sinh^{-1}(y) \). Specifically, the inverse hyperbolic sine uses a logarithmic expression to map the output back to \( x \). The transformation is given by the formula: \[ x = \sinh^{-1}(y) = \ln(y + \sqrt{y^2 + 1}) \] This expression ties together hyperbolic functions and logarithms in a beautiful way, providing a straightforward calculus tool to revert exponentiation in the hyperbolic context.

• \( \ln \) refers to the natural logarithm, a core mathematical function used extensively in analysis and complex calculations.
• The square root within the formula helps in handling the additional term that differentiates hyperbolic from typical trigonometric calculus.

Understanding these relationships enables the solving of complex exponential and hyperbolic equations by using properties of logarithms effectively.