Problem 42 - Contemporary Precalculus

Question

Let \(f(x)=\left\\{\begin{aligned} 1 / x & \text { if } x \neq 0 \\ 2 & \text { if } x=0 \end{aligned}\right.\) Find the domain of \(f\)

Step-by-Step Solution

Answer

Answer: The domain of \(f(x)\) is all real numbers, which can be represented as \(\{-\infty, +\infty\}\).

Step 1: Identify the domains of each rule

For the first rule, \(f(x) = \frac{1}{x}\), we need to determine when this expression makes sense. Notice that the only possible issue here is when \(x=0\), since division by zero is undefined. So, the domain of this rule is all real numbers except \(0\). Now, let's look at the second rule, \(f(x) = 2\). This expression is defined for every \(x\)-value since there are no restrictions on \(x\) when calculating \(f(x)\). However, we should keep in mind that the first rule covered all real numbers except \(0\), so the second rule serves only as an auxiliary function that covers the point \(x=0\).

Step 2: Combine the domains of each rule

Now, we need to combine the domain of each rule to find the domain of the piecewise-defined function. Since the first rule covers all real numbers except \(0\), and the second rule covers \(x=0\), we can conclude that the domain of \(f(x)\) includes all real numbers. The function itself is defined as the following:\(f(x)=\left\\{\begin{aligned} 1 / x & \text { if } x \neq 0 \\\ 2 & \text { if } x=0 \end{aligned}\right.\)

Step 3: Write the final answer

As we have determined in the previous steps, the domain of \(f(x)\) is all real numbers. In mathematical notation, we would write the domain of \(f(x)\) as: Domain of \(f(x) = \{-\infty, +\infty\}\). So, the final answer for the domain of \(f(x)\) is: Domain of \(f(x) = \{-\infty, +\infty\}\).

Key Concepts

Piecewise Function

A piecewise function is a type of function defined by different expressions or "pieces," depending on the input values. This means that the function behaves differently based on certain conditions, usually involving the variable.

In the piecewise function given in our problem, we have two distinct parts:

• For all values except zero, the function is defined as \( f(x) = \frac{1}{x} \). Here, \( x eq 0 \).
• For \( x = 0 \), the function simply equals \( 2 \).

The rule for a piecewise function is to select the appropriate piece based on the input value of \( x \). Different conditions determine which equation to use, allowing the function to adapt its behavior at specific points or intervals. This adaptability makes piecewise functions especially useful in modeling real-world scenarios where abrupt changes occur.

Function Evaluation

Function evaluation is the process of finding the output of a function based on a given input. To evaluate a piecewise function like \( f(x) \), you need to consider which piece to select based on the condition that \( x \) satisfies.

Here's how to evaluate the given piecewise function:

• If you choose an \( x \) not equal to zero, use the equation \( f(x) = \frac{1}{x} \). For example, if \( x = 3 \), then \( f(3) = \frac{1}{3} \).
• If \( x = 0 \), use the second part of the function: \( f(x) = 2 \). When \( x = 0 \), the output is clearly \( 2 \).

In summary, always check which condition between the pieces is met by \( x \) before plugging it into the function. This selective process ensures the correct evaluation of each piece, providing accurate and meaningful results.

Division by Zero

Division by zero is a mathematical operation that fundamentally lacks meaning, leading to undefined expressions. When you divide by zero, you essentially attempt to distribute a quantity into zero parts, which is impossible.

In the context of the piecewise function \( f(x) = \frac{1}{x} \) if \( x eq 0 \), division by zero becomes an issue for \( x = 0 \). This situation invalidates the operation, as the function cannot produce a real number output when \( x \) is zero under this rule.

• To prevent dividing by zero, the piecewise function includes a specific action for \( x = 0 \), which instead assigns the value \( f(x) = 2 \). This added condition circumvents the undefined expression, making the function valid for all real numbers.

Understanding division by zero is crucial in mathematics to avoid errors in problem-solving and ensure the logical consistency of functions and operations.