Problem 3 - College Algebra Essentials

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 x=4.

Step 1: Identify the function type

The function given, \(g(x)=\frac{3}{x-4}\), is a rational function because it is a ratio of two polynomials.

Step 2: Understanding domain restrictions on a rational function

Rational functions will be undefined when the denominator equals zero. This is because division by zero is undefined in mathematics.

Step 3: Identify when the function is undefined

To find when the function is undefined, you set the denominator equal to zero and solve for \(x\). In this case, setting \(x-4=0\), we get the solution \(x=4\). This means the function is undefined when \(x=4\).

Step 4: Find the domain

So, to find the domain of the function, we need all real numbers except where the function is undefined. Hence, the domain of the function is 'All real x, \(x \ne 4\)'.