Problem 61 - College Algebra with Modeling and Visualization

Question

Find the domain of the function. Write your answer in set-builder notation. $$ f(x)=\frac{1}{x^{2}-5} $$

Step-by-Step Solution

Answer

The domain is \( \{ x \in \mathbb{R} \mid x \neq \sqrt{5}, x \neq -\sqrt{5} \} \).

Step 1: Understanding the Function

The function given is \( f(x) = \frac{1}{x^2 - 5} \). It is a rational function, which means it is a fraction with a variable in the denominator. The domain of a rational function is all real numbers except those that make the denominator zero, as division by zero is undefined.

Step 2: Setting the Denominator Equal to Zero

To find the values that make the denominator zero, we set \( x^2 - 5 = 0 \). Solving this equation will show which values of \( x \) are not in the domain.

Step 3: Solving for Zero

Solve the equation \( x^2 - 5 = 0 \). Add 5 to both sides to get \( x^2 = 5 \). Then, take the square root of both sides, yielding \( x = \sqrt{5} \) and \( x = -\sqrt{5} \). These are the values where the function is undefined.

Step 4: Writing the Domain in Set-Builder Notation

The domain of the function is all real numbers except \( x = \sqrt{5} \) and \( x = -\sqrt{5} \). Using set-builder notation, this is written as \( \{ x \in \mathbb{R} \mid x eq \sqrt{5}, x eq -\sqrt{5} \} \).

Key Concepts

Rational Functions

Rational functions are an intriguing type of mathematical expression. They are essentially ratios of two polynomials. In simpler terms, a rational function is a fraction where both the numerator and the denominator are polynomials. For example, in the function \( f(x) = \frac{1}{x^2 - 5} \), the numerator is simply the constant 1, and the denominator is \( x^2 - 5 \).
Rational functions have special properties, particularly concerning their domains. The critical aspect to consider is the denominator. If the denominator becomes zero, the function is undefined. Therefore, finding the domain of a rational function includes identifying which values of \( x \) make the denominator zero and excluding them from the domain. This is because dividing by zero is mathematically impossible and undefined.
To handle rational functions effectively, remember:

• Always check what makes the denominator zero.
• Exclude these values from the domain.
• Present the domain using an appropriate notation, like set-builder notation.

Set-Builder Notation

Set-builder notation is a mathematical tool used to describe the elements of a set in a precise way. It's ideal for expressing the domain of functions where specific values need to be excluded, such as those found in rational functions. When dealing with a situation like \( f(x) = \frac{1}{x^2 - 5} \), where certain values make the function undefined, set-builder notation comes in handy to clearly list those exceptions.
The format of set-builder notation typically includes:

• A variable (e.g., \( x \)) that represents the elements in the set.
• A description or condition for these elements (e.g., "\( x eq \sqrt{5}, x eq -\sqrt{5} \)").
• The symbol \( \in \mathbb{R} \) indicating that \( x \) belongs to the set of all real numbers.

For example, the domain of the function \( f(x) = \frac{1}{x^2 - 5} \) can be expressed using set-builder notation as \( \{ x \in \mathbb{R} \mid x eq \sqrt{5}, x eq -\sqrt{5} \} \). This succinctly defines the domain as all real numbers except where \( x = \sqrt{5} \) and \( x = -\sqrt{5} \).
Using set-builder notation is not only a neat way to represent domains but also facilitates clear communication in mathematics, making it easier to understand and apply.

Undefined Values

Undefined values in mathematics, especially with functions, can be tricky. They occur when a function's expression doesn't result in a real number. In rational functions, like \( f(x) = \frac{1}{x^2 - 5} \), these undefined values arise when the denominator equals zero.
Mathematically, division by zero is a no-go, creating situations that are termed 'undefined'. To identify these values, set the denominator equal to zero and solve the equation. For the function above, solving \( x^2 - 5 = 0 \) gives us \( x = \sqrt{5} \) and \( x = -\sqrt{5} \). These are the x-values that make the denominator zero, rendering the whole function undefined at those points.
When dealing with undefined values, keep in mind:

• They must be excluded from the function's domain.
• They are crucial in understanding the behavior of the function, especially in graphing or calculus applications.

Understanding and dealing with undefined values is essential for a solid grasp of rational functions. It helps predict how the function behaves and ensures mathematical expressions are always defined for use.