Problem 9 - Elementary and Intermediate Algebra

Question

Explain why \(-4\) is not in the domain of \(f(x)=\frac{1}{x+4}\).

Step-by-Step Solution

Answer

-4 makes the denominator zero, so it's not in the domain.

Step 1: Understanding the Function

We need to determine when the function \( f(x) = \frac{1}{x+4} \) is undefined. A function is undefined when the denominator equals zero, as division by zero is not possible. Let's identify the value of \( x \) that makes \( x+4 = 0 \).

Step 2: Setting the Denominator to Zero

To find when the function is undefined, set the denominator equal to zero: \( x + 4 = 0 \). This equation will help us find the value of \( x \) that should not be included in the domain.

Step 3: Solving the Equation

Solve the equation \( x+4 = 0 \) to find the critical value. Subtract 4 from both sides: \( x = -4 \).

Step 4: Identifying Non-Domain Values

The solution \( x = -4 \) indicates that if \( x = -4 \), the denominator becomes zero, hence the function is undefined. Therefore, \(-4\) is not included in the function's domain.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually represented by \(x\)) for which the function is defined.
When we talk about the domain, we're concerned with identifying the values of \(x\) that can be safely plugged into the function without leading to undefined operations.
In simpler terms, if a function has a limitation, such as division by zero or taking the square root of a negative number, these will impact the domain.

• The domain includes all real numbers except those that lead to any non-permissible operation.
• For example, in the function \(f(x)=\frac{1}{x+4}\), we need to ensure that the denominator is not zero, because division by zero is not allowed.

Undefined Values

Undefined values are numbers, which when used in place of \(x\) in a function, lead to expressions that cannot be evaluated.
This typically happens when the function includes operations like division by zero or other undefined operations in real number arithmetic.

• To identify undefined values in a given function, look for input values that make the mathematical operation impossible.
• In our function \(f(x)=\frac{1}{x+4}\), the function is undefined when \(x+4=0\). Solving this equation \(x=-4\) reveals the undefined value.

Understanding which values make a function undefined helps us in defining the proper domain.

Division by Zero

Division by zero is a mathematical operation that is undefined in arithmetic.
It occurs when any number is divided by zero, which results in a value that mathematics cannot precisely define or represent.
The basic rule is that any such division is non-permissible.

• For instance, in our function \(f(x)=\frac{1}{x+4}\), we ensure that \(x+4\) is never zero because this would imply dividing by zero, which is undefined.
• This rule is crucial in constructing the function's domain, where any potential zero division must be excluded.

To avoid undefined results in functions, always check the denominator and ensure it doesn’t equal zero for any \(x\) value.