Sketching the Function f(x) x / √(x^2 - 9): A Comprehensive Guide

When sketches of mathematical functions are required, particularly those involving roots and fractions, it’s important to follow a systematic approach. In this article, we will guide you through the process of sketching the function f(x) x / √(x^2 - 9). This step-by-step approach will cover the domain, asymptotes, intercepts, and end behavior of the function.

1. Determining the Domain

The first step in the process of sketching a function is to determine its domain. The given function f(x) x / √(x^2 - 9) includes a square root, so the expression inside the square root must be non-negative. Therefore, we must solve the inequality: $$x^2 - 9 geq 0$$ This inequality is satisfied when x ≤ -3 or x ≥ 3. Thus, the domain of the function is given by the union of the intervals (-∞, -3] and [3, ∞).

2. Identifying Asymptotes

Vertical Asymptotes: A vertical asymptote occurs where the function is undefined because the denominator becomes zero. In this case, we need to solve the equation: $$x^2 - 9 0$$ Solving for x gives us x ±3. However, these values of x are not in the domain of the function, so there are no vertical asymptotes. Horizontal Asymptote: To determine the horizontal asymptote, we need to analyze the behavior of the function as x approaches ±∞. As x becomes very large (positive or negative), the term x / √(x^2 - 9) approaches 1 because the term x in the numerator dominates the term √(x^2 - 9). Therefore, the function approaches the horizontal asymptote y 1.

3. Determining Intercepts

x-intercepts: The x-intercepts occur where the function intersects the x-axis, meaning f(x) 0. Solving the equation x / √(x^2 - 9) 0 gives us x -1. However, this value is not in the domain of the function, so there are no x-intercepts. y-intercept: The y-intercept occurs when x 0. Evaluating the function at x 0, we get f(0) 0 / √(0^2 - 9) -1 / 3. Therefore, the y-intercept is at the point (0, -1/3).

4. End Behavior

As x approaches positive or negative infinity, the function f(x) x / √(x^2 - 9) approaches 1. This is because the term x in the numerator dominates the term √(x^2 - 9) as x becomes very large (positive or negative).

5. Plotting the Graph

Based on the above analysis, we can sketch the graph of f(x) x / √(x^2 - 9) as follows:

The domain of the function is (-∞, -3] ∪ [3, ∞). There are no vertical asymptotes. The function intersects the y-axis at (0, -1/3). There are no x-intercepts. The function approaches the horizontal asymptote y 1 as x approaches positive or negative infinity.

The shape of the graph would closely resemble an open curve, approaching the x-axis but never touching it. It would pass through the point (-1, 0) and approach the horizontal asymptote as x moves away from the origin. The graph would be symmetrical with respect to the y-axis due to the even powers of x in the numerator and denominator.

Conclusion

By following the systematic approach outlined above, we can accurately sketch the function f(x) x / √(x^2 - 9). Understanding the domain, asymptotes, intercepts, and end behavior is crucial for a comprehensive and accurate graph.