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The range of a function is the set of all possible output values related to the domain or input values. Specifically, when considering the hyperbolic sine function, denoted as \( \sinh x = \frac{e^x - e^{-x}}{2} \), the task is to determine what values \( \sinh x \) can attain. Other common examples of functions include linear functions or quadratic functions, which have their own unique ranges.
For \( \sinh x \), we seek to prove that any real number can be produced as an output of the function. This means that for each real number \( y \), there should exist a real number \( x \) such that \( y = \sinh x \).
The range, therefore, is all real numbers because no matter what number you think of, there will always be an \( x \) that results in that number when you calculate \( \sinh x \).

An inverse function essentially reverses the operation of the original function. For a function \( f(x) \), an inverse function \( f^{-1}(x) \) would satisfy the property \( f(f^{-1}(x)) = x \). In the context of hyperbolic sine, we want to find the inverse of \( \sinh x \).
To demonstrate the existence of an inverse function for \( \sinh x \), we solve the equation \( y = \sinh x \) for \( x \). This requires manipulating the equation such that \( x \) is expressed as a function of \( y \).
Through this process, we find \( x = \ln(y + \sqrt{y^2 + 1}) \), allowing us to rewrite \( x \) in terms of \( y \). This confirms that for any \( y \), there is a corresponding \( x \). Therefore, the inverse function of \( \sinh x \) does exist, further reinforcing that the range of \( \sinh x \) is all real numbers.

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \) and are solved using various techniques depending on the situation. When computing the inverse of \( \sinh x \), we arrive at a quadratic equation in terms of \( e^x \), given by \( e^{2x} - 2ye^x - 1 = 0 \).
The quadratic formula, \( e^x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a tool used to find the solutions for such equations. Here, with \( a = 1, b = -2y, \) and \( c = -1 \), we solve for \( e^x \), yielding \( e^x = y + \sqrt{y^2 + 1} \) because \( e^x \) must be positive.
Solving this quadratic confirms that no matter the value of \( y \), a real solution for \( x \) can be found, again demonstrating the comprehensive range of \( \sinh x \).

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). Natural logarithms are the inverse operations of the exponential function, meaning \( \ln(e^x) = x \).
When finding the inverse of \( \sinh x \), we employ the natural logarithm to solve for \( x \) from \( e^x = y + \sqrt{y^2 + 1} \). This step involves taking the natural logarithm of both sides, resulting in \( x = \ln(y + \sqrt{y^2 + 1}) \).
This application of the natural logarithm is crucial because it converts the exponential form back into a linear one, making \( x \) explicit in terms of \( y \). Understanding the natural logarithm is vital in functions involving growth and decay models, including hyperbolic functions like \( \sinh x \).