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Unveiling the Secrets of the Graph of x/(x-1): A Comprehensive Exploration

The graph of the function f(x) = x/(x-1) presents a fascinating study in mathematical behavior. Understanding its intricacies requires exploring its asymptotes, intercepts, domain, range, and overall shape. This comprehensive guide delves into each of these aspects, offering a detailed analysis suitable for students and enthusiasts alike. We'll unravel the secrets behind this seemingly simple function, revealing its surprising complexity and beauty. By the end, you'll not only be able to sketch the graph accurately but also understand the underlying mathematical principles that govern its form.

Introduction: A First Glance at x/(x-1)

At first sight, the function f(x) = x/(x-1) might seem straightforward. However, a closer examination reveals a rich tapestry of mathematical concepts. This rational function, defined as the ratio of two polynomials, exhibits characteristics not found in simpler linear or quadratic functions. We will explore its key features, including its vertical and horizontal asymptotes, x and y intercepts, and the behavior of the function as x approaches positive and negative infinity. Understanding these features allows us to accurately plot its graph and predict its behavior for any given input.

Finding the Vertical Asymptote: Where the Function Explodes

A vertical asymptote occurs where the denominator of a rational function equals zero, provided the numerator doesn't also equal zero at that point. In our function, f(x) = x/(x-1), the denominator is (x-1). Setting the denominator equal to zero gives us:

x - 1 = 0

Solving for x, we find x = 1. Therefore, the function has a vertical asymptote at x = 1. This means the graph approaches infinity or negative infinity as x gets closer and closer to 1, from either the left or the right. We will investigate this behavior more closely later.

Determining the Horizontal Asymptote: Long-Term Behavior

To find the horizontal asymptote, we examine the behavior of the function as x approaches positive and negative infinity. We can use the concept of limits:

lim (x→∞) x/(x-1)

To evaluate this limit, we can divide both the numerator and the denominator by x:

lim (x→∞) 1/(1 - 1/x)

As x approaches infinity, 1/x approaches zero. Therefore, the limit becomes:

lim (x→∞) 1/(1 - 0) = 1

Similarly, the limit as x approaches negative infinity is also 1:

lim (x→-∞) x/(x-1) = 1

This tells us that the function has a horizontal asymptote at y = 1. This means the graph approaches the line y = 1 as x becomes very large (positive or negative).

Identifying the x and y Intercepts: Where the Graph Crosses the Axes

The x-intercept is the point where the graph crosses the x-axis (where y = 0). To find this, we set f(x) = 0 and solve for x:

x/(x-1) = 0

This equation is satisfied only when the numerator is zero, which means x = 0. Therefore, the x-intercept is at (0, 0).

The y-intercept is the point where the graph crosses the y-axis (where x = 0). We can find this by substituting x = 0 into the function:

f(0) = 0/(0-1) = 0

Therefore, the y-intercept is also at (0, 0).

Domain and Range: Defining the Function's Territory

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since our function is undefined when the denominator is zero (x = 1), the domain is all real numbers except x = 1. We can express this in interval notation as (-∞, 1) ∪ (1, ∞).

The range of a function is the set of all possible output values (y-values). Because of the horizontal asymptote at y = 1 and the behavior of the function around the vertical asymptote, the range is all real numbers except y = 1. In interval notation, this is (-∞, 1) ∪ (1, ∞).

Analyzing the Behavior Around the Vertical Asymptote: Approaching Infinity

Let's examine the behavior of the function as x approaches the vertical asymptote at x = 1 from the left and from the right.

As x approaches 1 from the left (x → 1⁻): The numerator approaches 1, while the denominator approaches 0 from the negative side (a small negative number). Therefore, the function approaches negative infinity: lim (x→1⁻) x/(x-1) = -∞
As x approaches 1 from the right (x → 1⁺): The numerator approaches 1, while the denominator approaches 0 from the positive side (a small positive number). Therefore, the function approaches positive infinity: lim (x→1⁺) x/(x-1) = ∞

This behavior confirms the existence of the vertical asymptote and its direction.

Sketching the Graph: Bringing It All Together

Now that we have analyzed all the key features, we can sketch the graph of f(x) = x/(x-1). The graph will:

Pass through the origin (0,0).
Approach the vertical asymptote at x = 1 from both sides, going to negative infinity from the left and positive infinity from the right.
Approach the horizontal asymptote at y = 1 as x goes to positive or negative infinity.
Be a hyperbola-like curve, with branches in the quadrants II and IV.

The graph will clearly show the asymptotes, intercepts, and the overall behavior of the function. Remember to label all key points and asymptotes for clarity.

Further Exploration: Derivatives and Concavity

For a more in-depth analysis, we can examine the first and second derivatives of the function. The first derivative will reveal where the function is increasing or decreasing, and the second derivative will show the concavity (whether the graph is curving upwards or downwards). However, calculating and interpreting these derivatives adds a layer of complexity beyond the scope of a basic graphical analysis. This exploration would be suitable for more advanced mathematical studies.

Frequently Asked Questions (FAQ)

Q: Is the function f(x) = x/(x-1) continuous?

A: No, the function is discontinuous at x = 1 because of the vertical asymptote.

Q: Can the graph ever cross the horizontal asymptote?

A: In this specific case, no. The horizontal asymptote represents the long-term behavior of the function. Although a function can cross a horizontal asymptote, it typically does so only a finite number of times and not in the long run. A detailed analysis (often involving the derivative) would confirm this for this particular function.

Q: How does changing the numerator or denominator affect the graph?

A: Changing the numerator or denominator will alter the intercepts, asymptotes, and overall shape of the graph. For example, changing the numerator to 2x would stretch the graph vertically. Modifying the denominator would shift the vertical asymptote and change the behaviour around it. Exploring such variations allows for a deeper comprehension of how altering the function changes its graph.

Conclusion: A Journey Through a Rational Function

The seemingly simple function f(x) = x/(x-1) reveals a wealth of information about rational functions, asymptotes, limits, and graphical representations. By systematically investigating its key features, we have constructed a comprehensive understanding of its behavior. This journey emphasizes the importance of a step-by-step approach to analyzing mathematical functions, highlighting the interplay between algebraic manipulation, limit calculations, and graphical interpretation. This analysis provides a solid foundation for tackling more complex rational functions and further strengthens your understanding of fundamental mathematical concepts. The detailed exploration of this function serves as a valuable exercise in mathematical analysis and problem-solving.