Problem 16 - Precalculus Essentials

Question

Find the domain of each function. $$f(x)=\frac{1}{\frac{4}{x-2}-3}$$

Step-by-Step Solution

Answer

The domain of the function is all real numbers except \(x = \frac{10}{3}\).

Step 1: Set the denominator equal to zero

First, set the expression in the denominator equal to zero. So we have \(\frac{4}{x-2}-3 = 0\)

Step 2: Solve for x

Then solve this equation for x. Add 3 to both sides to get: \(\frac{4}{x-2}=3\). Multiply both sides by \(x-2\) to get: \(4 = 3(x -2)\). Then distribute the 3 and subtract 4 from both sides to isolate x. \(3x = 4 + 6 = 10\). Finally, divide by 3 to get: \(x = \frac{10}{3}\).

Step 3: Get the domain

Finally, the domain of the function is all real numbers except \(\frac{10}{3}\) since the function is undefined for \(x = \frac{10}{3}\).