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The domain of rational functions refers to all possible x-values that a function can take without leading to mathematically undefined situations. Specifically, for a function \[ f(x) = \frac{P(x)}{Q(x)} \]we need to determine when the denominator, \( Q(x) \), is not equal to zero. This is because division by zero is undefined. In our exercise, the denominator is \( x^2 + x \). **Steps to find the domain:**

• Factor the denominator: \( x(x + 1) = 0 \).
• Solve the equation: \( x = 0 \) and \( x = -1 \). These values make the denominator zero, hence they are excluded from the domain.

Thus, the domain of the function is all real numbers except \( x = 0 \) and \( x = -1 \). Knowing the domain is essential to understand where the function is defined.

Asymptotes are lines that the graph of a function approaches but never actually touches or crosses. For rational functions, these are typically vertical or horizontal. To locate them, we use:**Vertical Asymptotes:**- Vertical asymptotes occur where the denominator is zero, as long as the numerator is not zero at these points too. From our factorization:

• Vertical asymptotes are at \( x = 0 \) and \( x = -1 \).

**Horizontal Asymptotes:**- To determine horizontal asymptotes, compare the degrees of the numerator and denominator. If both have the same degree:

• The horizontal asymptote is the ratio of the leading coefficients. Our function has a horizontal asymptote at \( y = -2 \), because the leading coefficients of the numerator and denominator are \(-2\) and \(1\), respectively.

These asymptotes guide us in understanding the overall structure and end behavior of the function.

Intercepts are points where the graph of the function crosses the axes:**X-Intercepts:**- To find the x-intercepts, set the numerator equal to zero and solve for \( x \):

• \(-2x^2 + 10x - 12 = 0\). Factoring gives \(-2(x-3)(x-2) = 0\).
• The x-intercepts are \( x = 3 \) and \( x = 2 \).

**Y-Intercepts:**- The y-intercept is found by evaluating \( f(0) \), but in this case, since the denominator becomes zero at \( x=0 \), no y-intercept exists.Intercepts provide critical points through which the graph passes, helping to determine its precise shape and position.

Understanding the behavior of a graph involves examining how it behaves towards infinity and near asymptotes. This involves:**Analyzing Behavior Near Asymptotes:**- As we approach the vertical asymptotes at \( x = 0 \) and \( x = -1 \), the function can either head towards \( \infty \) or \(-\infty \). Evaluating points around these asymptotes helps determine the direction.**Behavior Between Intercepts:**- Check sample points in intervals divided by intercepts \( x = 2 \) and \( x = 3 \) to see if they lie above or below the horizontal asymptote \( y = -2 \).Such analysis gives insight into how the graph moves across its domain, whether it follows expected trends, and how it behaves near critical points like intercepts and asymptotes.