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Understanding the intercepts of a rational function is key to graphing and analyzing the function's behavior. The

• **x-intercepts** occur where the graph of the function crosses the x-axis. For this, the value of the function must be zero, meaning we only focus on the numerator. In the given function, \(-x^3-7x^2+16x+112=0,\)solving it approximately yields x-intercepts at (-8,0) and (4,0). These are the points where the graph intersects the x-axis.
• **y-intercepts** occur where the graph crosses the y-axis. This happens when the value of x is zero. Simply substitute 0 for x in the function. Evaluating the ratio gives \(f(0) = \frac{112}{28} = 4\),resulting in the y-intercept at (0,4).

Graphs of the function typically confirm these intercepts as visible crossing points on the axes. Being aware of where these intercepts lie helps in sketching the behavior of the function correctly.

Asymptotes represent lines that a graph approaches but never truly reaches. **Vertical asymptotes** occur when the denominator of a rational function is zero, creating undefined points unless the numerator can cancel these zeros. In \(f(x) = \frac{-x^{3}-7x^{2}+16x+112}{x^{2}+x+28},\)set \(x^{2} + x + 28 = 0.\)Here, attempts to find real roots result in complex solutions, meaning no real vertical asymptotes exist. This implies the graph continues unbroken in these regions. **Oblique asymptotes** arise when the degree of a polynomial in the numerator is exactly one more than the degree of the denominator. Performing polynomial division of \(-x^{3} -7x^{2} +16x +112\) by \(x^{2}+x+28\) yields \(-x-8,\)identifying an oblique asymptote at \(y = -x - 8.\)The graph gets closer and closer to this line indefinitely as x moves towards positive or negative infinity.

The domain and range describe all possible values input and output by the function.

• The **domain** of a rational function usually excludes values that create division by zero. Nevertheless, the denominator \(x^2 + x + 28\) lacks real roots, resulting in a domain of all real numbers \(-\infty < x < \infty\).
• The **range** involves analyzing the function's output behavior as x varies. The influence of an oblique asymptote \(y = -x - 8\) showcases general dominance of the polynomial's leading terms as x tends to infinities, implying the range also covers all real numbers \(-\infty < y < \infty\).

The broad domain and range demonstrate that the function extends without boundary over both axes, forming a complete view of its continuous nature.