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Intercepts are critical in understanding the behavior of a rational function. They tell us where the graph of the function crosses the axes.

**X-Intercepts**To find the x-intercepts of a rational function like \( r(x) = \frac{x^2 - x - 6}{x^2 + 3x} \), you must solve the equation when the output value is zero. This happens when the numerator is zero. In our example, solving \( x^2 - x - 6 = 0 \), we factor it to \( (x - 3)(x + 2) = 0 \). Thus, the roots are \( x = 3 \) and \( x = -2 \), which are our x-intercepts.

• Set the numerator of the function to zero.
• Solve for \( x \).
• The solutions are your x-intercepts.

**Y-Intercept**For the y-intercept, you evaluate the function at \( x = 0 \). However, if the function is undefined (as \( r(0) \) here), there is no y-intercept because it involves division by zero. Identifying whether a y-intercept exists is essential to mapping the graph accurately.

Asymptotes are lines that a graph approaches but never quite touches or crosses. Recognizing asymptotes helps to predict the behavior of a function as it moves towards infinity in any direction.

**Vertical Asymptotes**To find vertical asymptotes, identify where the denominator of the rational function equals zero. For our function, \( x^2 + 3x = 0 \), factor it to \( x(x + 3) = 0 \), which gives us \( x = 0 \) and \( x = -3 \). These values point to vertical asymptotes

• Set the denominator equal to zero.
• Solve for \( x \).
• Locations where the function becomes undefined are your vertical asymptotes.

**Horizontal Asymptotes**Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator. For \( r(x) \), both polynomials are degree 2, so the horizontal asymptote is determined by the ratio of the leading coefficients, which in this case results in \( y = 1 \).

• Compare the highest powers of the numerator and denominator.
• If they're equal in degree, the horizontal asymptote is \( y = \text{leading coefficient ratio} \).
• A horizontal line \( y = 1 \) acts as an approach line but isn't crossed by \( r(x) \).

Understanding the domain and range of a rational function is key to fully grasping where the function can exist and what y-values it can take.

**Domain**The domain of a function refers to all the possible x-values for which the function is defined. In rational functions, this excludes any x-values that make the denominator zero. For \( r(x) = \frac{x^2 - x - 6}{x^2 + 3x} \), the domain excludes \( x = 0 \) and \( x = -3 \) because they lead to a division by zero.

• Write the function's denominator.
• Set it equal to zero and solve for restricted x-values.
• Domain is all real numbers except these restricted values.

**Range**While domains involve input values, the range of a function involves all possible output values (y-values). The horizontal asymptote \( y = 1 \) suggests that the function will approach but never reach \( y = 1 \), while extending through other real numbers, giving this function a comprehensive range of real numbers. Understanding both domain and range provides a complete picture of the function's behavior in two dimensions.