Find z-values

Learning Objectives Define a z-value (zero) in the context of a rational function. Identify the numerator and denominator of a rational function to set up the problem. Solve for the z-values of a rational function by setting the numerator equal to zero. Verify that a potential z-value is not a removable discontinuity (a hole) or a vertical asymptote. Distinguish between z-values, vertical asymptotes, and holes by analyzing both the numerator and the denominator. Interpret z-values as the x-intercepts on the graph of a rational function. Ever wonder how engineers find the exact points where a complex system's output is zero, like finding the break-even point in a business model? 📈 Let's learn the core mathematical skill behind it! This tutorial focuses on a fundam...

TermDefinitionExample Rational FunctionA function that can be written as the ratio of two polynomial functions, P(x) and Q(x), in the form f(x) = P(x) / Q(x), where Q(x) is not the zero polynomial.f(x) = (x^2 - 4) / (x - 1) Z-Value (Zero or Root)A z-value of a function is an input value 'z' from the domain that results in an output of zero, i.e., f(z) = 0. Graphically, these are the x-intercepts of the function.For f(x) = x - 5, the z-value is 5 because f(5) = 5 - 5 = 0. NumeratorThe polynomial in the top part of the fraction of a rational function. The zeros of the numerator are the potential z-values of the function.In f(x) = (x^2 + 2x) / (x - 3), the numerator is P(x) = x^2 + 2x. DenominatorThe polynomial in the bottom part of the fraction of a rational function. The zeros of...

The Zero-Numerator Rule For a rational function f(x) = \frac{P(x)}{Q(x)}, the z-values are found by solving the equation P(x) = 0. A fraction can only be equal to zero if its numerator is zero. This is the primary step for finding all potential z-values. The Exclusion Principle A value 'z' is a true zero of f(x) = \frac{P(x)}{Q(x)} only if P(z) = 0 AND Q(z) \neq 0. After finding the zeros of the numerator, you must check if any of those values also make the denominator zero. If a value makes both zero, it corresponds to a hole in the graph, not a z-value (x-intercept).