Problem 5 - Algebra 2

Question

Find any points of discontinuity for each rational function. $$ y=\frac{x^{2}+5 x+6}{x^{2}+6 x+9} $$

Step-by-Step Solution

Answer

The function \(y=\frac{x^{2}+5x+6}{x^{2}+6x+9}\) is discontinuous at \(x = -3\).

Step 1: Identify the Denominator of the Function

The denominator of the given function is \(x^{2}+6x+9\). This is the part of the function that will determine any points of discontinuity.

Step 2: Set the Denominator Equal to Zero

We set the denominator equal to zero to find which values of x make the function undefined: \(x^{2}+6x+9 = 0\)

Step 3: Solve the Equation for x

Solve the equation for the variable \(x\). The equation is a quadratic equation and can be factored to \((x+3)^2 = 0\). When this equation is solved, the solution for \(x\) is \(-3\).

Step 4: Identify the Points of Discontinuity

The points of discontinuity of the function are the values of x where the function is undefined. With a solution of -3 from the previous step, it indicates that the function is discontinuous at \(x = -3\).

Key Concepts

Understanding Points of Discontinuity

Points of discontinuity occur in a function when there are certain values of the variable for which the function is not defined. In rational functions, this happens when the denominator is equal to zero. Since division by zero is undefined, these particular x-values cause the function to "break" or be discontinuous.

To find these points in a rational function, follow these simple steps:

• Identify the denominator.
• Set the denominator equal to zero.
• Solve for x to find the point(s) of discontinuity.

After solving the denominator for zero, any solutions provide the x-values where the function cannot exist. In our exercise, these solutions reveal that the rational function is discontinuous at certain x-values. Understanding these points helps make sense of why and where a rational function cannot be plotted along the coordinate plane.

Introduction to Rational Functions

Rational functions are fractions involving polynomials in the numerator and the denominator. A function becomes rational when it takes the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. These functions can have interesting behaviors such as asymptotes and discontinuities.

Since rational functions can be complex, it's crucial to examine them carefully:

• Look at both numerator and denominator.
• Check for factors that might cancel each other.
• Determine where the function is undefined.

Analyzing a rational function tells you about the curve's graph, such as where it approaches infinity (vertical asymptotes) or where it has breaks (discontinuities). In practical applications, understanding this can help when sketching graphs or solving real-world problems involving rates or ratios.

Solving Quadratic Equations

The techniques used to solve quadratic equations are essential in finding the points of discontinuity in rational functions. A quadratic equation generally looks like \( ax^2 + bx + c = 0 \), and can often be factored or solved using the quadratic formula.

In our problem:

• The quadratic we solved was \( x^2 + 6x + 9 = 0 \).
• This can be factored into \((x+3)^2 = 0\).
• Solving this gives a repeated solution: \( x = -3 \).

Recognizing when a quadratic can be easily factored streamlines the process of identifying discontinuities in rational equations. A single solution such as this indicates a point where the function completely breaks down.