Problem 79 - College Algebra with Corequisite Support

Question

For the following exercises, identify the removable discontinuity. $$ f(x)=\frac{x^{3}+x^{2}}{x+1} $$

Step-by-Step Solution

Answer

x = -1 is a removable discontinuity.

Step 1: Find Points of Discontinuity

To identify removable discontinuities in the function \( f(x) = \frac{x^3 + x^2}{x+1} \), first determine the points where the function is undefined. Since the denominator is \( x+1 \), the function is undefined at \( x = -1 \).

Step 2: Simplify the Function

Next, simplify \( f(x) \) to see if the function can be redefined at the discontinuity to make it continuous. Factor the numerator as \( x^2(x + 1) \). Thus, the function becomes \( f(x) = \frac{x^2(x + 1)}{x+1} \). Now, cancel the \( x+1 \) terms in the numerator and the denominator.

Step 3: Factor and Cancel Common Terms

The expression simplifies to \( f(x) = x^2 \), except where \( x = -1 \). The division by zero at \( x = -1 \) was the only obstruction. Since \( x = -1 \) was not in the domain before cancellation, \( x = -1 \) is a removable discontinuity.

Key Concepts

Discontinuity in Functions

A function is discontinuous when you can't draw it without lifting your pen off the paper. Discontinuities can occur in various forms, such as jumps, infinite discontinuities, or removable discontinuities. Removable discontinuities occur when a point on the graph is undefined, but the limit exists. This means that by redefining the function at that point, the discontinuity can "be removed." In the given exercise, the function \( f(x) = \frac{x^3 + x^2}{x+1} \) is initially undefined at \( x = -1 \) due to division by zero. However, upon analyzing the expression further, you can often correct this by simplifying and canceling out terms that create the discontinuity.

Simplifying Rational Expressions

Simplifying rational expressions involves rewriting a fraction in its simplest form. This usually requires factoring both the numerator and the denominator to identify and cancel common factors. The goal is to simplify the expression so that any removable discontinuities become apparent. In the exercise, the given function \( f(x) = \frac{x^3 + x^2}{x+1} \) is simplified by factoring the numerator as \( x^2(x+1) \). Once factored, you observe that the \( x + 1 \) term can be canceled from both the numerator and the denominator, leading to a simplified function \( f(x) = x^2 \). This process reveals the removable discontinuity at \( x = -1 \).

Factorization in Algebra

Factorization is an essential step in Algebra when dealing with expressions that require simplification. It involves breaking down a complex expression into simpler factors that are multiplied together. In the context of rational expressions, factorization helps to simplify the expression and potentially cancel terms. The original function \( f(x) = \frac{x^3 + x^2}{x+1} \) demonstrates this process well. The numerator \( x^3 + x^2 \) is factored to \( x^2(x+1) \). Recognizing this factorization allows the identification of the \( x+1 \) term, which cancels out with the denominator, simplifying the whole expression. Factorization is a powerful tool for revealing underlying properties of functions, such as removable discontinuities.