Question

h(x) = \frac{4}{x - 5}
determine for each x-value whether it is in the domain of h or not.

in domain not in domain
0 \circ \circ
4 \circ \circ
5 \circ \circ

Explanation:

Step1: Recall domain of rational function

A rational function \( h(x)=\frac{f(x)}{g(x)} \) has domain all real numbers except where \( g(x) = 0 \). For \( h(x)=\frac{4}{x - 5} \), set denominator \( x-5=0 \), solve \( x = 5 \). So domain is all real numbers except \( x = 5 \).

Step2: Check \( x = 0 \)

Substitute \( x = 0 \) into denominator: \( 0 - 5=-5
eq0 \). So \( x = 0 \) is in domain.

Step3: Check \( x = 4 \)

Substitute \( x = 4 \) into denominator: \( 4 - 5=-1
eq0 \). So \( x = 4 \) is in domain.

Step4: Check \( x = 5 \)

Substitute \( x = 5 \) into denominator: \( 5 - 5 = 0 \). So \( x = 5 \) is not in domain.

Answer:

• For \( x = 0 \): In domain
• For \( x = 4 \): In domain
• For \( x = 5 \): Not in domain

(To mark: Circle "In domain" for \( 0 \) and \( 4 \), circle "Not in domain" for \( 5 \))