{Use of Tech} Let f(x) = ln((x+1)/(x-1)) and g(x) = ln ((x+1)/(x-1)).
b. Sketch graphs of f and g to show that these functions do not differ by a constant.

Verified step by step guidance

1. Identify the functions: Both f(x) and g(x) are given as f(x) = ln((x+1)/(x-1)) and g(x) = ln((x+1)/(x-1)). Notice that they are identical, which suggests they should be the same function.
2. Understand the concept: If two functions differ by a constant, their graphs will be vertically shifted versions of each other. This means that for all x in their domain, f(x) = g(x) + C, where C is a constant.
3. Determine the domain: The domain of both functions is x > 1 and x < -1, since the argument of the logarithm, (x+1)/(x-1), must be positive.
4. Sketch the graph: Plot the graph of f(x) = ln((x+1)/(x-1)) over its domain. Since f(x) and g(x) are identical, their graphs will overlap completely, indicating they do not differ by a constant.
5. Conclude from the graph: Since the graphs of f(x) and g(x) are identical and overlap completely, it confirms that they do not differ by a constant. If they did, one graph would be a vertical shift of the other.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental function in calculus, particularly in relation to growth rates and areas under curves. Understanding the properties of the natural logarithm, such as its domain and range, is essential for analyzing functions like f(x) and g(x).

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize their behavior. For functions f(x) and g(x), sketching their graphs helps to identify key features such as intercepts, asymptotes, and overall shape. This visual representation is crucial for determining whether two functions differ by a constant, as it allows for direct comparison of their outputs across the same input values.

Difference of Functions

The difference of two functions, f(x) and g(x), is expressed as f(x) - g(x). If this difference is a constant for all x in the domain, it indicates that the two functions are parallel and differ only by that constant. In the context of the given functions, analyzing their difference will reveal whether they maintain a consistent vertical shift or if they diverge in behavior.