Graphing the Function f(x) log?(x 1): Asymptotes, Intercepts, and Key Features

Understanding how to sketch the graph of a logarithmic function can be a valuable skill in mathematics. In this article, we will explore the function f(x) log?(x 1) in detail, including its key features such as asymptotes, intercepts, and slopes. This will help you visualize and interpret the behavior of the function.

Understanding the Function

The function f(x) log?(x 1) is a logarithmic function, which means that it can be interpreted as the exponent to which the base 2 must be raised to get (x 1). Understanding the behavior of such functions is crucial for many applications, including data analysis, signal processing, and various fields of engineering.

Key Features of f(x) log?(x 1)

Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches but never touches. For f(x) log?(x 1), the vertical asymptote occurs where the function is undefined. This happens at the point where the argument of the logarithm, x 1, is zero. Therefore, the vertical asymptote is:

x -1

At this point, the function is undefined, and as x approaches -1 from the right, the value of f(x) approaches negative infinity.

Horizontal Asymptote

A horizontal asymptote is a line that the graph approaches as x tends towards positive or negative infinity. For the function f(x) log?(x 1), the horizontal asymptote does not exist because log?(x 1) increases without bound as x increases. Therefore, as x approaches positive infinity, f(x) approaches positive infinity.

Intercepts

y-intercept: The y-intercept is the point where the graph intersects the y-axis. This occurs when x 0. Substituting x 0 into the function, we get:

[f(0) log?(0 1) log?(1) 0]

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

x-intercept: The x-intercept is the point where the graph intersects the x-axis. This occurs when f(x) 0. Setting log?(x 1) 0, we solve for x:

[log?(x 1) 0 Rightarrow 2^0 x 1 Rightarrow x -1]

Thus, the x-intercept is at the point (-1, 0).

Grasping the Behavior of the Function

To fully understand the behavior of the function, we can plot some key points and analyze the slopes at certain points on the graph.

Slope at the X-axis Intercept

The slope of the tangent line at the x-intercept can be calculated using the derivative of the function. The derivative of f(x) log?(x 1) is:

[f'(x) frac{1}{(x 1) ln(2)}]

At the x-intercept, where x -1, the slope is:

[f'(-1) frac{1}{ln(2)} approx 1.443]

This means that the tangent line at the x-intercept makes an angle of approximately 55.77° with the positive x-axis.

Slope at the Y-axis Intercept

The slope of the tangent line at the y-axis intercept (0, 0) can also be calculated using the derivative. The slope is:

[f'(0) frac{1}{ln(2)} approx 1.443]

This indicates that the tangent line at the y-axis intercept makes an angle of approximately 55.77° with the positive x-axis as well.

Conclusion

Understanding the key features of the function f(x) log?(x 1) through vertical and horizontal asymptotes, intercepts, and slopes is essential for visualizing and interpreting its behavior. By plotting these points and analyzing the slopes, we can gain a deeper understanding of how logarithmic functions behave.

Keywords: logarithmic function, vertical asymptote, horizontal asymptote