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X-intercepts are the points where a graph crosses the x-axis. For the rational function \(f(x) = \frac{-x^3 - 7x^2 + 16x + 112}{x^2 + x + 28}\), we find the x-intercepts by setting the numerator equal to zero. We are essentially looking for the values of \(x\) that make the whole expression zero. This can often involve solving a polynomial equation, and sometimes requires the use of tools like synthetic division or a calculator for complex polynomials.In this function, calculating \(-x^3 - 7x^2 + 16x + 112 = 0\) yields three solutions: \(x = -8, x = 2,\) and \(x = -7\). These solutions mean that the x-intercepts of this function are \((-8, 0)\), \((2, 0)\), and \((-7, 0)\). Every zero of the numerator contributes to an x-intercept, dictating where the graph will touch or cross the x-axis.

The y-intercept of a graph is the point where it crosses the y-axis. This intersects when \(x = 0\). To find the y-intercept of a rational function, simply substitute \(x = 0\) into the equation and solve for \(f(x)\).For our function \(f(x) = \frac{-x^3 - 7x^2 + 16x + 112}{x^2 + x + 28}\), plugging in \(x = 0\) yields:\[f(0) = \frac{0^3 - 7(0)^2 + 16(0) + 112}{0^2 + 0 + 28} = \frac{112}{28} = 4\]Thus, the y-intercept is at the point \((0, 4)\). This is where the graph will touch or cross the y-axis, indicating the value of \(f(x)\) when \(x\) is zero, giving us a straightforward point on the graph.

An oblique asymptote occurs in a rational function when the degree of the numerator is one higher than the degree of the denominator. It represents the behavior of the function as \(x\) approaches infinity or negative infinity.For the function \(f(x) = \frac{-x^3 - 7x^2 + 16x + 112}{x^2 + x + 28}\), the numerator is of degree 3, and the denominator is of degree 2. This indicates the presence of an oblique asymptote.To find the equation of the oblique asymptote, perform polynomial long division of the numerator by the denominator. The quotient will provide the equation. In this case, dividing \(-x^3 - 7x^2 + 16x + 112\) by \(x^2 + x + 28\) gives us the oblique asymptote equation: \[y = -x - 8\]This oblique asymptote describes the slanted line that the function graph approaches, offering insight into the end behavior of the graph.

The domain of a rational function consists of all the x-values for which the function is defined. This is generally all real numbers except where the denominator equals zero. For the function \(f(x) = \frac{-x^3 - 7x^2 + 16x + 112}{x^2 + x + 28}\), solving the equation \(x^2 + x + 28 = 0\) shows no real solutions since the discriminant is negative. Thus, the denominator never equals zero, and the domain includes all real numbers.The range of a rational function is the set of possible y-values. Since this function has no vertical asymptotes and features an oblique asymptote, the function approaches \(-x - 8\) as \(x\) tends toward infinity in either direction. Consequently, the function spans all real numbers vertically, indicating that the range is also all real numbers.Understanding domain and range helps visualize how the graph behaves horizontally and vertically, providing critical insight into the overall aspect of the function on a coordinate plane.