Questions: Find the domain of the function. f(x) = 6/(x-2) + 7/(x-5) The domain of f(x) is

Transcript text: Find the domain of the function. \[ f(x)=\frac{6}{x-2}+\frac{7}{x-5} \] The domain of $f(x)$ is $\square$

Solution

Solution Steps

To find the domain of the function $ f(x) = \frac{6}{x-2} + \frac{7}{x-5} $, we need to determine the values of $ x $ for which the function is defined. The function is undefined where the denominators are zero. Therefore, we need to find the values of $ x $ that make $ x-2 $ and $ x-5 $ equal to zero and exclude these values from the domain.

Solution Approach

Identify the values of $ x $ that make the denominators zero.
Exclude these values from the set of all real numbers to determine the domain.

Step 1: Identify Points of Discontinuity

The function $ f(x) = \frac{6}{x-2} + \frac{7}{x-5} $ is undefined where the denominators are zero. We find the points of discontinuity by solving the equations: \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

Step 2: Determine the Domain

The domain of the function consists of all real numbers except the points where the function is undefined. Therefore, we exclude $ x = 2 $ and $ x = 5 $ from the set of real numbers.