Find the domain of the function [tex]f(x)=\frac{2 x^2+16 x+4}{2 x+4}[/tex]. What is the only value of [tex]x[/tex] not in the domain?
[tex]x=[/tex]
Answer (1)
The domain of a rational function excludes values that make the denominator zero.
Set the denominator 2 x + 4 equal to zero.
Solve for x : 2 x + 4 = 0 ⟹ x = − 2 .
The only value not in the domain is − 2 .
Explanation
Finding the Domain We are given the function f ( x ) = 2 x + 4 2 x 2 + 16 x + 4 and asked to find its domain. The domain of a rational function consists of all real numbers except for values of x that make the denominator equal to zero. To find these values, we set the denominator equal to zero and solve for x .
Setting the Denominator to Zero We set the denominator equal to zero: 2 x + 4 = 0
Solving for x To solve for x , we subtract 4 from both sides of the equation: 2 x = − 4 Then, we divide both sides by 2: x = 2 − 4 = − 2
The Value Not in the Domain Therefore, the only value of x that is not in the domain of the function is x = − 2 .
Examples
Consider a scenario where you are distributing resources, and the function represents the amount of resource each person gets. The domain restriction means there's a certain number of people for which the distribution is not defined (e.g., dividing by zero). Understanding the domain helps you avoid such scenarios, ensuring a valid and fair distribution.
