Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=(x²-16)/(x+4)

Verified step by step guidance

1. Identify the given rational function: \(f(x) = \frac{x^2 - 16}{x + 4}\).
2. Recognize that the numerator \(x^2 - 16\) is a difference of squares, which can be factored as \(x^2 - 16 = (x - 4)(x + 4)\).
3. Rewrite the function using the factored form: \(f(x) = \frac{(x - 4)(x + 4)}{x + 4}\).
4. Simplify the expression by canceling the common factor \((x + 4)\) in numerator and denominator, but note that \(x \neq -4\) because division by zero is undefined. The simplified form is \(f(x) = x - 4\), with a restriction on the domain.
5. Match the function to the description that corresponds to a linear function with a hole (removable discontinuity) at \(x = -4\) due to the canceled factor.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the form and behavior of rational functions is essential for analyzing their properties such as domain, asymptotes, and simplification.

Simplification and Factorization

Simplifying rational functions involves factoring polynomials in the numerator and denominator to cancel common factors. This process can reveal removable discontinuities (holes) and simplify the function to a more recognizable form.

Domain and Discontinuities

The domain of a rational function excludes values that make the denominator zero. Identifying these values helps determine vertical asymptotes or holes, which are points where the function is undefined or discontinuous.