Problem 10 - Calculus for Biology and Medicine

Question

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\frac{1}{x^{2}-1} $$

Step-by-Step Solution

Answer

The function \( f(x) = \frac{1}{x^2 - 1} \) is discontinuous at \( x = 1 \) and \( x = -1 \).

Step 1: Identify the Denominator

First, we need to identify what causes the denominator in the function to become zero since division by zero leads to discontinuity. The function given is \( f(x) = \frac{1}{x^2 - 1} \), which means we focus on the expression \( x^2 - 1 \).

Step 2: Set the Denominator to Zero

Set the expression in the denominator equal to zero and solve for \( x \): \[ x^2 - 1 = 0 \]

Step 3: Solve for x

Rewrite the equation as \( x^2 = 1 \) and solve for \( x \) by taking the square root of both sides. \[ x = \pm 1 \]

Step 4: Verify Discontinuity at the Solutions

The points where \( x = 1 \) and \( x = -1 \) make the denominator zero, which means \( f(x) \) is undefined at these points and hence discontinuous.

Key Concepts

Division by Zero

In mathematics, division by zero is a concept that arises when one attempts to divide a number by zero. This operation is undefined because dividing any number by zero does not result in a finite or meaningful number. For example, consider dividing 5 by zero, which we can mathematically express as \( \frac{5}{0} \). No number exists that you can multiply by 0 to get 5, which creates a problem called a 'definition gap'.

In the context of functions, when the denominator of a function equals zero, the function becomes undefined due to division by zero. This often leads to discontinuities in the graph of the function. That is why it is crucial to identify where in the function's expression division by zero might occur, so we can determine any points of discontinuity.

It's important to remember that not all functions are impacted the same way by division by zero. Each requires individual assessment based on its algebraic expression.

Solving Equations

Solving equations involves finding the values of variables that make an equation true. This is an essential skill in algebra and is particularly useful when analyzing function behavior or identifying points of discontinuity.

Consider the function given: \( f(x) = \frac{1}{x^2 - 1} \). To find where this function is discontinuous, we need to solve the equation where the denominator is zero. This means solving \( x^2 - 1 = 0 \).

To solve for \( x \), identify what value makes the equation true by isolating \( x^2 \) on one side, resulting in \( x^2 = 1 \). The next step is to solve for \( x \) by taking the square root of both sides, yielding \( x = \pm 1 \). These values are critical, as they define where the function is undefined and discontinuous.

Undefined Expressions

Undefined expressions occur in mathematics when an expression does not result in a clearly defined numerical value. A common cause of undefined expressions is division by zero, where the denominator of a fraction equals zero.

Returning to the function \( f(x) = \frac{1}{x^2 - 1} \), notice that at \( x = 1 \) and \( x = -1 \), the denominator becomes zero, resulting in the expression being undefined. Identifying such points is crucial because it indicates where a function lacks a real output, which translates to discontinuities on the graph.

This undefined nature means at these points, the function 'breaks' or is no longer continuous, impacting its graph and limiting its ability to be used meaningfully within that context. Recognizing points of undefined expressions helps in understanding and analyzing functions more holistically.