Problem 13 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{4}{\frac{3}{x}-1}\) is \( (-\infty, 3) \cup (3, \infty) \).

Step 1: Identify the Zero-Denominator Condition

Because dividing by zero is undefined, we must make sure that the denominator does not equal zero. Therefore, we must find the value of x that will make the denominator equal zero, i.e., solve the equation \(\frac{3}{x}-1 = 0\).

Step 2: Solve the Equation

The equation from Step 1 can be solved as follows: add 1 to both sides to get \(\frac{3}{x} = 1\), then cross-multiply to isolate x, resulting in \(x = 3\).

Step 3: State the Domain of the Function by excluding the Zero-Denominator Value

The domain of the function includes all real numbers except for the value that makes the denominator equal zero. In this case, x is all real numbers except 3. Therefore, the domain of h(x) can be written in interval notation as\( (-\infty, 3) \cup (3, \infty) \).