Problem 53 - College Algebra

Question

Find the domain of each function. $$g(x)=\frac{3}{x-4}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{3}{x-4}\) is all real numbers except 4.

Step 1: Identify the denominator

The denominator of the function \(g(x) = \frac{3}{x-4}\) is \(x - 4\). This is what will be used to find the domain.

Step 2: Set the denominator equal to zero

Setting the denominator \(x - 4\) equal to zero and solving the resultant equation will give the values of x where the function is undefined. This is achieved as follows:\[x - 4 = 0\]Solving the equation yields \(x = 4\). Thus, \(x = 4\) is where the function is undefined.

Step 3: Formulate the domain

The domain of any function consists of the set of all real numbers except those that make the function undefined. Here, the function is undefined at \(x = 4\). Hence, the domain of the function \(g(x) = \frac{3}{x-4}\) are all real numbers except 4.