Question

The domain of the function \(f(x)=\frac{3 x-6}{x-4}\) is _______.

Short answer

The domain is all real numbers except x = 4.

Step-by-step solution

- Understand the Function

First, identify the form of the function given. The function is a rational function: \[f(x) = \frac{3x - 6}{x - 4}\]

- Identify Restrictions

Rational functions have restrictions where the denominator cannot be zero. Find the values of x that make the denominator zero: \[x - 4 = 0\]

- Solve for Restrictions

Solve the equation from Step 2: \[x - 4 = 0\]\[x = 4\]

- State the Domain

The domain of the function includes all real numbers except the value that makes the denominator zero (x = 4). Therefore, the domain is all real numbers except x = 4: \[\text{Domain: } x eq 4\]

Key concepts

Rational Functions

Rational functions are fractions where the numerator and denominator are both polynomials. For example, \( f(x) = \frac{3x - 6}{x - 4} \) is a rational function because both '3x - 6' and 'x - 4' are polynomials. Understanding the components of rational functions is essential for working through problems involving their domain. The key feature of these functions is that they can be undefined where their denominator is zero. For example, in \( f(x) = \frac{3x - 6}{x - 4} \), the function will be undefined if \( x - 4 \) equals zero.

Denominator Restrictions

When working with rational functions, it is crucial to identify 'denominator restrictions' or the values of \( x \) that make the denominator zero. These values make the function undefined. For instance, with \( f(x) = \frac{3x - 6}{x - 4} \), you set the denominator \( x - 4 \) to zero and solve the equation \( x - 4 = 0 \). Solving this equation, you get: \( x = 4 \). Therefore, x = 4 is a restriction and should be excluded from the domain of the function.

Solving Equations

Solving equations to find restrictions on the domain is straightforward. To prevent the function from becoming undefined, the steps are:

• Set the denominator equal to zero.
• Solve for \( x \).
• Exclude the solution from the domain.

In our example, solve \( x - 4 = 0 \) to find \( x = 4 \). Thus, the domain of the function \( f(x) = \frac{3x - 6}{x - 4} \) is all real numbers except x = 4. The final domain is written as \( x eq 4 \).