Problem 9 - Precalculus Essentials

Question

Find the domain of each function. $$f(x)=\frac{1}{x+7}+\frac{3}{x-9}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\frac{1}{x+7}+\frac{3}{x-9}\) is \(x \in \mathbb{R}\), \(x \neq -7\), \(x \neq 9\)

Step 1: Looking at the first fraction

The denominator of the first fraction is x+7. We can set this equal to zero and solve: \(x+7=0\). By subtracting 7 from both sides we get: \(x=-7\). So -7 is a value at which the given function is undefined.

Step 2: Looking at the second fraction

The denominator of the second fraction is x-9. We can set this equal to zero and solve: \(x-9=0\). By adding 9 to both sides we get: \(x=9\). So 9 is another value at which the given function is undefined.

Step 3: Formulating the domain

Having found the values at which the function is undefined, we can formulate the domain. The domain of a function is all the possible x-values. For this function, the domain is all real numbers except -7 and 9.