Question

Find the domain of the function. $$y=\frac{1}{3 x}$$

Short answer

The domain of the function \(y=\frac{1}{3 x}\) is all real numbers except x \(\neq\) 0.

Step-by-step solution

Identify the format of the function

This function is a simple reciprocal function, which follows the format \(y=\frac{1}{x}\). This function has a vertical asymptote at x = 0. This means that, for the function \(y=\frac{1}{3 x}\), the graph approaches y = 0 as x approaches positive or negative infinity, but the graph never touches or crosses the y-axis.

State restrictions for the denominator

Since division by zero is undefined, we express the domain by excluding x values for which the denominator of the fraction equals zero. In this function, the denominator is 3x. So, setting the denominator equal to zero, we have: 3x = 0. Solving this equation for x, we get x = 0.

Exclude the restrictive x values from the domain of the function

As we determined in step 2, we cannot use x = 0 as an input into the function, because it would make the denominator equal to zero and the function is undetermined. Therefore, the domain of the function \(\frac{1}{3 x}\) is all real numbers except x \(\neq\) 0.