Question

select the correct answer.
what is the domain of this function?
$f(x) = -\frac{1}{4}|2x + 1| - 1$
a. $(-infty, -\frac{1}{4}$
b. $(-infty, -\frac{1}{2}$
c. $(-infty, infty)$
d. $\frac{1}{4}, infty)$

Explanation:

Step1: Recall the domain of absolute value functions

The function given is \( f(x)=-\frac{1}{4}|2x + 1|-1 \). The absolute value function \( |u| \), where \( u = 2x+1 \), is defined for all real numbers \( x \) because we can substitute any real number for \( x \), and the expression inside the absolute value (the argument) will be a real number, and the absolute value of any real number is defined.

Step2: Determine the domain of the given function

Since the function \( f(x) \) is composed of an absolute value function (which is defined for all real numbers) and linear operations (multiplication by \( -\frac{1}{4} \) and subtraction of 1), there are no restrictions on the values of \( x \) that we can plug into the function. That is, \( x \) can be any real number. In interval notation, the set of all real numbers is \( (-\infty, \infty) \).

Answer:

C. \((-\infty, \infty)\)