Problem 78 - Algebra 2

Question

For each rational function, find any points of discontinuity. $$ y=\frac{2}{x+1} $$

Step-by-Step Solution

Answer

The point of discontinuity of the function \(y=\frac{2}{x+1}\) is at \(x = -1\).

Step 1: Set the denominator equal to zero

To find the points of discontinuity, set the denominator of the function equal to zero. Here, the function given is \(y=\frac{2}{x+1}\). Therefore, set \(x + 1 = 0\).

Step 2: Solve the equation

Solving the equation \(x + 1 = 0\) for \(x\), gets \(x = -1\).

Step 3: Determine the points of discontinuity

The solution of the equation from step 2 provides the points of discontinuity. Thus, the point of discontinuity for the given function is \(x = -1\).

Key Concepts

Points of Discontinuity

In mathematics, especially when dealing with rational functions, points of discontinuity are values of the variable for which the function is undefined. Think of them like roadblocks in the path of a function's journey. At these points, the function suddenly has a huge gap where it cannot be evaluated. Identifying these points is crucial for understanding the nature of the function.Discontinuity occurs in rational functions when the denominator of the fraction becomes zero. Why is this important? Because dividing by zero is undefined in mathematics. This makes the function have an infinite gap or a hole in its path at that particular value.For example, consider the function \(y = \frac{2}{x+1}\). Here, the function may have a point of discontinuity. To find it, we need to find when the denominator \(x + 1\) is zero. This discovery is essential in graphing the function and understanding its behavior as \(x\) approaches these critical points.

Setting Denominators to Zero

The process of identifying the points of discontinuity begins by setting the denominator of the rational function to zero. This is because discontinuity happens only when you can't divide by a number, which in mathematical terms means dividing by zero.Let's break it down with our function \(y = \frac{2}{x+1}\).

• The denominator is \(x + 1\).
• Set it equal to zero: \(x + 1 = 0\).

This equation will help us find the exact points that cause the function to become undefined. These are the values that you can't plug into your function since they would make the denominator zero, leading to mathematical mayhem!By setting the denominator to zero, we create an equation that, once solved, reveals exactly where the function "stops being friendly" and becomes discontinuous.

Solving Linear Equations

Once you've set the denominator to zero, the next step involves solving this simple linear equation. Linear equations form the basics of algebra, and solving them will help us find those points of discontinuity.Take the equation from our function: \(x + 1 = 0\). To solve it, you just need to isolate \(x\):

• Subtract 1 from both sides: \(x = -1\).

This result, \(x = -1\), is the exact point at which the function has a discontinuity.Solving linear equations in this context is often straightforward, but it's important because it allows us to understand the limitations of the function. Knowing these limitations helps when graphing the function and predicting its behavior around these pesky points of discontinuity.