Problem 2 - College Algebra

Question

Find the domain of each rational function. $$f(x)=\frac{7 x}{x-8}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{7x}{x-8}\) is all real numbers except \(x = 8\).

Step 1: Identify the Rational Function

First, identify the given function. In this case it is a rational function of the form \(f(x) = \frac{7x}{x-8}\). A rational function is a ratio of two polynomials, and in this problem, 7x and (x-8) are the polynomials in the numerator and the denominator, respectively.

Step 2: Set the Denominator Equal to Zero

To find the values that are not included in the domain of the rational function, set the denominator equal to zero and solve for x. In this case, setting \(x-8 = 0\), we find that \(x = 8\). Since the denominator of a fraction cannot be zero, \(x = 8\) is excluded from the domain because it makes the denominator of our rational function zero.

Step 3: Write the Final Answer for the Domain

Now that we know which value to exclude, we clearly state the domain of the function. The domain of this function is \(x\) such that \(x\) is a real number except \(x ≠ 8\).