Graph. f(x)=(x^2+5 x+6)/(x+3)

Name: Problem 33, Chapter 0: Calculus and Its Applications
Uploaded: 2020-10-22T15:07:56-08:00
Duration: 1 min 37 s
Channel: Linh Vu
Description: Graph. f(x)=(x^2+5 x+6)/(x+3)

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials. Understanding rational functions involves knowing that they may be simplified, have restricted domains, and can exhibit asymptotic behavior or discontinuities depending on the factors present in the numerator and denominator.

Domain Restrictions

For any rational function, it is important to determine the domain by identifying values that make the denominator zero, as these lead to undefined expressions. These restrictions may affect the overall behavior and continuity of the function.

Factoring Polynomials

Factoring is a key algebraic technique used to simplify expressions. In the context of rational functions, factoring the numerator or denominator (when possible) helps in revealing common factors that can be canceled, which simplifies the expression and highlights potential discontinuities.

Removable Discontinuities (Holes)

When a common factor exists in both the numerator and the denominator of a rational function and is canceled out, it creates a removable discontinuity, or hole, at that specific value. Although the simplified form of the function does not explicitly show this, the original domain still excludes the hole's location.

Graphing Techniques for Simplified Functions

After simplifying the rational function by canceling common factors, the graph of the function can often be interpreted as that of a simpler function (such as a linear or polynomial function) with adjustments for the holes. Graphing involves plotting the function while clearly indicating the discontinuities or asymptotic behaviors introduced by the original expression.

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