Problem 14 - College Algebra Essentials

Question

Find the domain of each function. $$h(x)=\frac{5}{\frac{4}{x}-1}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{5}{\frac{4}{x}-1}\) is all real numbers except 0 and 4, or in other words, \(x≠0, 4\).

Step 1: Identify the Restrictions

Identify restrictions for the function based on fraction and reciprocal rules. For all fractions, values of \(x\) that would set the denominator equal to zero are not allowed. Also, for all reciprocal functions in the form \(1/x\), \(x\) cannot equal zero. So for this function \(h(x)=\frac{5}{\frac{4}{x}-1}\), \(x\) cannot be zero due to the reciprocal function, and the expression \(\frac{4}{x}-1\) in the denominator cannot equal zero.

Step 2: Solve for the Denominator equal to Zero

Solve the restriction \(\frac{4}{x}-1=0\) for \(x\). First, add 1 to both sides to get \(\frac{4}{x}=1\). Then, divide both sides by 4 to get \(1/x = 1/4\). Finally, to solve for \(x\), invert both sides of the equation to get \(x=4\). Therefore, \(x\) cannot be 4 since this would make the denominator of the function equal to zero.

Step 3: Compile the Domain

Combine the findings from steps 1 and 2 to set up the domain of the function. The domain is all real numbers except for the values identified in steps 1 and 2, which are \(x=0\) and \(x=4\). Therefore, the domain of \(h(x)=\frac{5}{\frac{4}{x}-1}\) is \(x≠0, 4\).