Graph Of X 1 X 1

Decoding the Graph of y = 1/(x-1) + 1: A Comprehensive Analysis

The function y = 1/(x-1) + 1, a simple yet revealing example from rational functions, presents a rich landscape for exploration. Understanding its graph requires a blend of algebraic manipulation, conceptual grasp of asymptotes, and the ability to visualize transformations. This article will delve deep into the intricacies of this function, exploring its key features, behaviors, and the methods used to accurately graph it.

Understanding the Building Blocks: Rational Functions and Transformations

Before we dive into the specifics of y = 1/(x-1) + 1, let's refresh our understanding of its constituent parts. The function is a rational function – a function that can be expressed as the ratio of two polynomials. In this case, the numerator is 1, and the denominator is (x-1).

The graph of the basic rational function, y = 1/x, is a hyperbola with two branches. One branch resides in the first quadrant (x > 0, y > 0), while the other is in the third quadrant (x < 0, y < 0). Both branches approach but never touch the x and y axes, which are its asymptotes.

Now, let's consider the transformations applied to y = 1/x to obtain y = 1/(x-1) + 1:

• Horizontal Shift: The term (x-1) in the denominator represents a horizontal shift to the right by one unit. This means the entire graph of y = 1/x is moved one unit to the right.

• Vertical Shift: The "+1" outside the fraction represents a vertical shift upward by one unit. This further translates the shifted graph one unit upwards.

Identifying Key Features: Asymptotes, Intercepts, and Domain/Range

Understanding the key features of a function is critical for accurately graphing it. Let’s analyze these features for y = 1/(x-1) + 1:

1. Vertical Asymptote

A vertical asymptote occurs where the denominator of a rational function is equal to zero and the numerator is not zero. In this case, the denominator is (x-1), which equals zero when x = 1. Therefore, the vertical asymptote is the line x = 1. This line represents a boundary that the graph approaches but never crosses.

2. Horizontal Asymptote

A horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, we consider the limit of the function as x approaches infinity:

lim (x→∞) [1/(x-1) + 1] = 1

Similarly, as x approaches negative infinity:

lim (x→-∞) [1/(x-1) + 1] = 1

This means the horizontal asymptote is the line y = 1. The graph will approach this line as x moves towards positive or negative infinity.

3. x-intercept(s)

The x-intercept(s) are the points where the graph intersects the x-axis (i.e., where y = 0). To find them, we set y = 0 and solve for x:

0 = 1/(x-1) + 1

-1 = 1/(x-1)

-(x-1) = 1

-x + 1 = 1

x = 0

Therefore, the x-intercept is at (0, 0).

4. y-intercept

The y-intercept is the point where the graph intersects the y-axis (i.e., where x = 0). To find it, we substitute x = 0 into the equation:

y = 1/(0-1) + 1 = -1 + 1 = 0

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

5. Domain and Range

The domain of a function represents all possible x-values, while the range encompasses all possible y-values.

• Domain: The function is undefined when the denominator is zero (x = 1). Therefore, the domain is all real numbers except x = 1, which can be expressed as (-∞, 1) U (1, ∞).

• Range: Because of the horizontal asymptote at y = 1, the function will never actually reach the value of 1. The range is therefore all real numbers except y = 1, which can be written as (-∞, 1) U (1, ∞).

Graphing the Function: A Step-by-Step Approach

Now, equipped with the knowledge of asymptotes, intercepts, and domain/range, we can accurately sketch the graph.

1. Draw the Asymptotes: Start by drawing the vertical asymptote (x = 1) and the horizontal asymptote (y = 1) as dashed lines on the coordinate plane.

2. Plot the Intercepts: Plot the x-intercept and y-intercept at (0, 0).

3. Consider the Branches: The graph will have two branches, one on each side of the vertical asymptote.

4. Sketch the Branches: Knowing that the function approaches the asymptotes but never touches them, sketch the branches accordingly. The branch to the left of the vertical asymptote (x < 1) will approach the asymptotes in the second and third quadrants. The branch to the right of the vertical asymptote (x > 1) will approach the asymptotes in the first and fourth quadrants.

Advanced Analysis: Derivatives and Concavity

For a more in-depth analysis, we can explore the function's first and second derivatives to understand its rate of change and concavity.

First Derivative:

The first derivative, f'(x), indicates the slope of the tangent line at any point on the graph. Finding the derivative of y = 1/(x-1) + 1 using the power rule and chain rule, we get:

f'(x) = -1/(x-1)²

Notice that f'(x) is always negative. This confirms that the function is always decreasing on its domain.

Second Derivative:

The second derivative, f''(x), provides information about the concavity of the function. Differentiating f'(x), we have:

f''(x) = 2/(x-1)³

The second derivative is positive when x > 1 and negative when x < 1. This tells us that the function is concave up for x > 1 and concave down for x < 1. There is no inflection point because the function is discontinuous at x = 1.

Applications and Real-World Connections

Rational functions, like y = 1/(x-1) + 1, model various real-world phenomena. They can represent:

• Inverse relationships: Where one variable increases, the other decreases (but not proportionally).

• Physical limitations: Asymptotes can model limitations in systems. For example, the function could describe the relationship between the speed of a vehicle and the amount of time it takes to reach a destination, where the asymptote represents the physical limitations of speed.

• Economic Models: In economics, rational functions might model the relationship between supply and demand, with asymptotes representing market saturation or resource constraints.

Conclusion: A Complete Picture of y = 1/(x-1) + 1

Through a detailed analysis of its asymptotes, intercepts, domain/range, derivatives, and concavity, we've gained a complete understanding of the graph of y = 1/(x-1) + 1. This simple rational function serves as a powerful example illustrating the importance of understanding transformations, asymptotes, and the interplay between algebraic manipulation and graphical representation. By carefully considering these factors, we can confidently sketch accurate and informative graphs of rational functions, extending our understanding to more complex scenarios and real-world applications. This thorough understanding allows for a deeper appreciation of the behavior and significance of such functions in various fields.