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Derivatives are a fundamental concept in calculus. They help us understand how a function changes at any given point. Essentially, a derivative tells us the slope of a function's graph at a particular point. For a basic function like a line, the slope is constant, but for more complex functions, the slope can change. For the function given in the exercise, \(f(x) = 1 + x + \frac{1}{x}\), we calculate the derivative, \(f'(x)\), to find how this function’s rate of change behaves. By using rules of differentiation: - The derivative of a constant like 1 is 0. - The derivative of \(x\) is 1.- The derivative of \(\frac{1}{x}\) can be determined using the power rule, leading to \(-\frac{1}{x^2}\). Putting it all together, we derive that \(f'(x) = 1 - \frac{1}{x^2}\). This derivative function helps us understand where the original function is increasing or decreasing.

Graphing functions is a key skill in visually understanding how they behave. Plotting a function can highlight important features such as intercepts, slopes, and curvatures. For the function \(f(x) = 1 + x + \frac{1}{x}\), graphing it with a calculator allows us to observe these features directly. You will see how the function behaves close to critical points and asymptotes. When graphing, here are some steps to follow:

• Identify key points and asymptotes that might exist.
• Determine the function’s domain and range to know where it is defined.
• Use a calculator or graphing software for plotting the equation accurately.

By conducting this visual analysis, you can predict where the function crosses the x-axis or has steep changes, which are important for understanding its overall behavior.

A reciprocal function is any function of the form \( \frac{1}{x} \). These functions are interesting because they exhibit behavior that can be quite different from typical linear or quadratic functions. For the function \(f(x) = 1 + x + \frac{1}{x}\), \( \frac{1}{x} \) is the reciprocal part. Reciprocal functions have a characteristic hyperbola shape and possess vertical asymptotes where the function is undefined, typically where \(x=0\). Some properties of reciprocal functions include:

• They have vertical asymptotes at \(x = 0\).
• Their graph is symmetric with respect to the origin for negative and positive values.
• As \(x\) approaches infinity, the value of \( \frac{1}{x} \) approaches 0.

Understanding these behaviors is crucial when analyzing functions that have \( \frac{1}{x} \) components, as these intricacies greatly affect the function’s overall behavior.

Asymptotes are lines that a graph of a function approaches but never touches. They are critical in studying functions as they indicate boundaries of behavior. Asymptotes can be vertical, horizontal, or slant. In the function \(f(x) = 1 + x + \frac{1}{x}\), the term \(\frac{1}{x}\) creates a vertical asymptote at \(x = 0\), since the function becomes undefined when \(x\) is 0. This tells us that the graph will have a gap or immense change at this point.Characteristics of asymptotes:

• Vertical asymptotes occur where the function's denominator is zero, e.g., \(x = 0\) for \(\frac{1}{x}\).
• Horizontal asymptotes describe the behavior as \(x\) approaches infinity. For functions like \( \frac{1}{x} \), the asymptote is \(y = 0\).

By understanding asymptotes, students can better grasp the limits of the function's graph, identify potential discontinuities, and make informed predictions about function behavior.