Problem 16 - College Algebra 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 this function is all real numbers except \( x=\frac{10}{3} \)

Step 1: Identify the Denominator

First, identify the denominator of the function. It is necessary to set the denominator not equal to zero. Here, the denominator is \( \frac{4}{x-2}-3 \)

Step 2: Set the Denominator not Equal to Zero

To find values that are not in the domain of the function, set the denominator not equal to zero and solve for \( x \):\(\frac{4}{x-2}-3 \neq 0\)

Step 3: Isolate the Fraction

Add 3 to both sides of the equation:\(\frac{4}{x-2} \neq 3\)

Step 4: Solve for x

Finally, find the x-value that would cause the denominator to be zero. Multiply both sides of the equation by \( x - 2 \):\(4 \neq 3x - 6\)Then, add 6 to both sides and divide all terms by 3:\(x \neq \frac{10}{3}\)