Problem 67 - Calculus Early Transcendentals

Question

More graphing Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way. $$f(x)=\frac{x \sqrt{\left|x^{2}-1\right|}}{x^{4}+1}$$

Step-by-Step Solution

Answer

Short Answer: The complete graph of the function $$f(x)=\frac{x \sqrt{\left|x^{2}-1\right|}}{x^{4}+1}$$ has the following properties: 1. Domain: (-∞, +∞) and Range: (-∞, +∞) 2. X-intercepts: x = -1 and x = 1; Y-intercept: (0,0) 3. Vertical Asymptotes: None; Horizontal Asymptote: y = 0 4. Symmetry: Symmetric with respect to the origin (odd function) 5. Plotting key points and using a graphing utility confirms that the graph converges to 0 as x approaches infinity and minus infinity, and verifies the symmetry with respect to the origin. By including these properties, we obtain a sketch of the complete graph of the given function \(f(x)\).

Step 1: Identify the Domain and Range

The domain of a function consists of all valid x-values for which the function is defined. The function $$f(x)=\frac{x \sqrt{\left|x^{2}-1\right|}}{x^{4}+1}$$ is well-defined for all x-values except when the denominator is equal to zero. In this case, the denominator $$x^{4}+1$$ always gives a positive value (strictly greater than 0) for all real x. Thus, the domain of the given function is (-∞, +∞). Now, let's find the range which consists of all possible y-values the function can take. As the function has a rational form, we know that the function will have a horizontal asymptote. We can quickly analyze this by taking the limit of the function when x approaches infinity and minus infinity. $$\lim_{x \to \infty} f(x) = 0$$ $$\lim_{x \to -\infty} f(x) = 0$$ Hence, the function converges to 0 at extreme x-values. Therefore, the range of $$f(x)$$ lies between $$(-\infty, +\infty)$$

Step 2: Identify X and Y Intercepts

An x-intercept is the point where the function touches or crosses the x-axis (y=0), and a y-intercept is the point where the function touches or crosses the y-axis (x=0). - X-intercept: $$f(x) = 0$$ We can observe that when $$x^2 - 1 = 0$$, the function will be equal to zero. Thus, the x-intercepts occur at x = -1 and x = 1. - Y-intercept: $$f(0)$$ By substituting x = 0 in the given function, we get: $$f(0) = \frac{0 \sqrt{\left|-1\right|}}{1} = 0$$ So, the y-intercept occurs at the point (0,0).

Step 3: Identify Vertical and Horizontal Asymptotes

Vertical asymptotes occur when the denominator of the rational function is equal to zero, and the function tends to infinity. In this case, the denominator $$x^{4}+1$$ is always greater than zero for all x, so there are no vertical asymptotes. Horizontal asymptotes occur when the function approaches a constant value as x tends towards infinity. We already found that the function converges to 0 at extreme x-values. Therefore, the horizontal asymptote is the line y=0.

Step 4: Identify Symmetry

To check for symmetry, we can analyze whether the function is even or odd. - Even function: $$f(-x) = f(x)$$ $$f(-x) = \frac{-x \sqrt{\left|(-x)^{2}-1\right|}}{(-x)^{4}+1} = -\frac{x \sqrt{\left|x^{2}-1\right|}}{x^{4}+1} \neq f(x)$$ - Odd function: $$f(-x) = -f(x)$$ $$f(-x) = -f(x)$$ holds true So, the function is symmetric with respect to the origin.

Step 5: Plot Key Points and the Complete Graph

Now we will plot the key points identified above, including the x and y intercepts, and the horizontal asymptote. We can then use a graphing utility to confirm that the graph converges to 0 as x approaches infinity and minus infinity. Also, verify the symmetry of the graph with respect to the origin. By combining the results from all the steps, we can sketch the complete graph of the given function \(f(x)\).

Key Concepts

Domain and Range of Functions

Understanding the domain and range of functions is crucial when graphing rational functions. The domain is the set of all possible inputs (x-values) for which the function is defined. For the function \(f(x)=\frac{x \sqrt{\left|x^{2}-1\right|}}{x^{4}+1}\), the domain is all real numbers because the denominator \(x^{4}+1\) is never zero, and square roots are defined for non-negative numbers.

The range, on the other hand, consists of all possible outputs (y-values) that the function can produce. Even though the function is defined for all real numbers, its range is affected by its behavior at extreme values of x. By determining the limits as x approaches infinity, we found that the function approaches zero. This suggests that the function does not have any bounds in its output, making the range of the function also all real numbers, \( (-\infty, +\infty) \).

X-Intercepts and Y-Intercepts

The x-intercepts and y-intercepts are points where the graph of a function crosses the axes. The x-intercepts occur when the output y is zero. For the given function, this happens when \(x^{2} - 1 = 0\), which gives us x-intercepts at x = -1 and x = 1. Traditionally, to find the x-intercepts, you would set the numerator equal to zero; however, due to the absolute value in our function, we need to consider the points where the value inside the square root is zero.

The y-intercept is found by evaluating the function at x = 0. For our function, this calculation yields \(f(0) = 0\), indicating that the y-intercept is at the origin (0,0). Identifying intercepts helps us sketch the function with more accuracy.

Horizontal and Vertical Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. The concept of vertical asymptotes comes into play when the denominator of a rational function approaches zero. In such cases, the function's value tends towards infinity. However, \(x^{4}+1\) cannot be zero for any real x, so the given function has no vertical asymptotes.

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. If the function levels out to a constant value, that value is the horizontal asymptote. For our function, as x grows indefinitely in either direction, the function approaches the horizontal asymptote y=0. This means the graph flattens out at extreme x-values.

Symmetry of Functions

The symmetry of a function can greatly simplify the graphing process. When a function is even, it is symmetric about the y-axis, which means \(f(-x) = f(x)\). Conversely, an odd function is symmetric about the origin, satisfying \(f(-x) = -f(x)\). For our exercise, by testing the function \(f(-x)\) against \(f(x)\), it's clear that the function is odd, as it meets the latter condition. This symmetry ensures that once we've plotted the function for positive values of x, we can reflect those points through the origin to find the corresponding points for negative x-values, allowing for an efficient graphing approach.