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Rational functions are expressions that involve the division of two polynomials. In simpler terms, a rational function has the form: \[ f(x) = \frac{P(x)}{Q(x)} \] where * \(P(x)\) and \(Q(x)\) are polynomials * \(Q(x)\) is not equal to zero The degrees of these polynomials can be used to analyze various characteristics of the function, such as its intercepts, asymptotes, and end behavior. Remember, a rational function can take many complex forms. Even though they might appear intimidating, breaking them down step-by-step makes them more approachable and easier to understand.

The y-intercept of a function is the point where the graph intersects the y-axis. This occurs at x = 0. To find the y-intercept for any function, substitute x = 0 into the function and solve for y. In this exercise, the function is: \[ f(x) = \frac{x^4 - 3x^3 - 21x^2 + 43x + 60}{x^4 - 6x^3 + x^2 + 24x - 20} \] Plugging in x = 0 simplifies the function significantly. All terms containing x will be zero, leaving: \[ f(0) = \frac{60}{-20} \rightarrow f(0) = -3 \] This means the y-intercept of the function is -3. Thus, the graph of the function intersects the y-axis at the point (0, -3).

Function evaluation involves finding the value of a function for a given input. When evaluating a function, simply replace the variable with the given input and perform the operations. Here's the process using the provided function: 1. Begin with the rational function: \[ f(x) = \frac{x^4 - 3x^3 - 21x^2 + 43x + 60}{x^4 - 6x^3 + x^2 + 24x - 20} \] 2. To find the y-intercept, set \(x = 0\): \[ f(0) = \frac{0^4 - 3(0)^3 - 21(0)^2 + 43(0) + 60}{0^4 - 6(0)^3 + (0)^2 + 24(0) - 20} = \frac{60}{-20} \] 3. Simplify the fraction to get: \[ f(0) = -3 \] This procedure can be used to evaluate many types of functions, helping to identify key characteristics like intercepts, critical points, and more. Function evaluation is a fundamental skill in algebra and calculus, and practice will make it second nature.