Problem 107 - Precalculus

Question

The hyperbolic sine function is defined by \(\sinh x=\frac{e^{x}-e^{-x}}{2}\) Find its inverse function \(\sinh ^{-1} x\).

Step-by-Step Solution

Answer

The inverse function is \( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \).

Step 1: Understanding the Inverse Function

For a function defined by \( y = f(x) \), its inverse function \( x = f^{-1}(y) \) is obtained by solving the equation for \( x \) in terms of \( y \). In this case, we will find \( x \) such that \( y = \sinh(x) = \frac{e^x - e^{-x}}{2} \).

Step 2: Setting Up the Equation

Start by setting \( y = \frac{e^x - e^{-x}}{2} \) to find \( x \) in terms of \( y \). This gives us the equation to solve: \( y = \frac{e^x - e^{-x}}{2} \).

Step 3: Clear the Fraction

Multiply both sides of the equation by 2 to eliminate the fraction: \[ 2y = e^x - e^{-x} \]

Step 4: Multiply Through by \( e^x \)

Multiply the entire equation by \( e^x \) to get rid of the \( e^{-x} \) term: \[ 2y e^x = e^{2x} - 1 \]

Step 5: Rearrange to Quadratic Form

Rearrange the equation to the standard quadratic form: \[ e^{2x} - 2ye^x - 1 = 0 \] This is a quadratic equation in terms of \( e^x \).

Step 6: Solve the Quadratic Equation

Use the quadratic formula \( e^x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve the equation, where \( a = 1 \), \( b = -2y \), and \( c = -1 \): \[ e^x = \frac{2y \pm \sqrt{(2y)^2 + 4}}{2} \] Simplifying, we find: \[ e^x = y \pm \sqrt{y^2 + 1} \]

Step 7: Select the Correct Root

Since \( e^x \) must be positive, choose the positive root: \[ e^x = y + \sqrt{y^2 + 1} \]

Step 8: Solve for \( x \)

Take the natural logarithm of both sides to solve for \( x \): \( x = \ln(y + \sqrt{y^2 + 1}) \)

Step 9: Express the Inverse Function

Therefore, the inverse hyperbolic sine function is:\( \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \)

Key Concepts

Hyperbolic Sine Function

The hyperbolic sine function is similar to the familiar sine function, but it is derived from exponential functions rather than circular functions. It is widely used in mathematics, particularly in calculus and complex analysis.
The hyperbolic sine function is written as \( \sinh x \), and it's defined as:

• \( \sinh x = \dfrac{e^x - e^{-x}}{2} \)

This equation means that for any value of \( x \), \( \sinh x \) is a specific combination of exponential growth and decay.

• The term \( e^x \) represents exponential growth, while \( e^{-x} \) represents exponential decay.

The function itself is not periodic like the trigonometric sine, but it exhibits certain symmetries, such as being an odd function \( (\sinh(-x) = -\sinh x) \). Understanding how \( \sinh x \) behaves is crucial for mastering hyperbolic functions and solving relevant mathematical problems.

Solving Quadratic Equations

Quadratic equations are integral to many mathematical applications. In general, a quadratic equation takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. To solve a quadratic equation, especially when it's derived from another function such as the hyperbolic sine, the quadratic formula is often used.

• The quadratic formula is: \( x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Applying this to the equation we encountered in the original exercise, which was \( e^{2x} - 2ye^x - 1 = 0 \), we see:

• \( a = 1 \), \( b = -2y \), and \( c = -1 \)
• Using these values, we solve the equation for \( e^x \)

Only the positive root is valid because exponential functions must remain positive. This root enables us to find the inverse of the hyperbolic sine function.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental concept in mathematics. It is the inverse of the exponential function \( e^x \). The natural logarithm has a base of the mathematical constant \( e \), which is approximately 2.718.
In our step-by-step solution, we encountered the natural log when solving for \( x \) after finding \( e^x \). The equation \( x = \ln(y + \sqrt{y^2 + 1}) \) shows the conversion from the exponential form back into a logarithmic form.

• This conversion is possible because, in logarithms, exponentiation and logarithms are inverse operations.

The natural log is vital in calculus because it's the only logarithm whose derivative simplifies directly to \( 1/x \). It also greatly helps in solving equations involving exponentials, making it a key tool in mathematical analysis and problem-solving.

Exponential Functions

Exponential functions are expressions where the variable appears in the exponent, such as \( e^x \). They exhibit rapid growth or decay, depending on the sign of the variable.
The fundamental nature of exponential functions is shown in their constant ratio of change, making them essential in fields like biology, finance, and physics.
In the original exercise, exponential functions played a key role in defining the hyperbolic sine function.

• Both \( e^x \) and \( e^{-x} \) determine the values of \( \sinh x \).

The inverse hyperbolic sine function leverages these properties to express \( x \) in terms of \( \ln(y + \sqrt{y^2 + 1}) \), showing how exponentials substitute and transform into logarithmic expressions. Understanding exponential functions provides a deep insight into their integrals, derivatives, and their applications across a wide range of mathematical and natural phenomena.