Problem 3 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Type

The function \( g(x) = \frac{3}{x-4} \) is a rational function, which is a function of the form \( f(x) = \frac{p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomial functions and \( q(x) \) is not zero.

Step 2: Find the Exclusions

The function is undefined when the denominator equals zero because division by zero is undefined in mathematics. Therefore, solve the equation \( x-4 = 0 \). This gives \( x = 4 \). Therefore, \( 4 \) is not included in the function's domain.

Step 3: Write Out the Domain

The domain of function \( g(x) \) is all real numbers except for \( x = 4 \). This can be written in interval notation as \( (-\infty, 4) \cup (4, +\infty) \).