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Vertical asymptotes occur where the denominator of a rational function equals zero and makes the function undefined. To find vertical asymptotes, we need to factor the denominator and solve for when it equals zero. Given the function $$f(x) = \frac{2x^2 + 1}{x^3 - x}$$First, set the denominator equal to zero:$$x^3 - x = 0$$We factor this to$$x(x^2 - 1) = 0$$and further to$$x(x - 1)(x + 1) = 0$$This gives solutions at$$x = 0$$,$$x = 1$$, and$$x = -1$$. Each of these solutions corresponds to a vertical asymptote. Therefore, the vertical asymptotes are at $$x = 0$$,$$x = 1$$,and $$x = -1$$.

Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator in a rational function. To find horizontal asymptotes: 1. Note the degrees of the numerator and the denominator. 2. Compare these degrees to determine the appropriate rule.For the function $$f(x) = \frac{2x^2 + 1}{x^3 - x}$$, the degree of the numerator $$2x^2 + 1$$ is 2,and the degree of the denominator $$x^3 - x$$ is 3. Since the degree of the numerator (2) is less than the degree of the denominator (3), the horizontal asymptote is at $$y = 0$$. When the degree of the numerator is smaller, the horizontal asymptote will always be at$$y = 0$$.

Intercepts are the points where the function crosses the axes. There are two types: x-intercepts and y-intercepts.**x-Intercepts**: These occur where the numerator of the rational function equals zero, provided the denominator does not equal zero.Given the function $$f(x) = \frac{2x^2 + 1}{x^3 - x}$$,we set the numerator to zero:$$2x^2 + 1 = 0$$This gives$$2x^2 = -1$$, but you will notice that a negative number cannot equal a square result for real numbers. Therefore, there are no real x-intercepts for this function.**y-Intercepts**: These occur where the function crosses the y-axis. To find it, evaluate the function at $$x = 0$$:$$f(0) = \frac{2(0)^2 + 1}{(0)^3 - 0} = \frac{1}{0}$$The value is undefined, so there is no y-intercept for this function.

Rational functions are expressions of the form$$\frac{P(x)}{Q(x)}$$,where both$$P(x)$$ and$$Q(x)$$ are polynomials. For more complex rational functions like $$f(x) = \frac{2x^2 + 1}{x^3 - x}$$, key characteristics include: - **Asymptotes**: These are found where the function's denominator equals zero (vertical) and by comparing degrees of the numerator and denominator (horizontal). - **Intercepts**: Determine where the function intersects the x-axis and y-axis.Understanding these key features of rational functions helps in sketching their graphs and analyzing their behavior.

Graphing functions like $$f(x) = \frac{2x^2 + 1}{x^3 - x}$$ requires understanding its key features:1. **Vertical Asymptotes**: Found at $$x = 0$$,$$x = 1$$,and $$x = -1$$. These lines illustrate where the graph approaches but never touches.2. **Horizontal Asymptote**: Found at $$y = 0$$,and it shows the behavior of the function as $$x \to \text{infinity or -infinity}$$.3. **Intercepts**: In this function, there are no x-intercepts or y-intercepts.To sketch, plot the asymptotes first. Then, consider the behavior of the function near these asymptotes and the origin. Use additional points for more precision, but recognizing asymptotes and intercepts will give a strong initial outline.