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Chapter 1: Problem 3

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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 \( (-\infty, 4) \cup (4, +\infty) \).

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