To determine the domain of the function [tex]\( f(x) = -\frac{1}{4} |2x + 1| - 1 \)[/tex], we need to identify all the possible values of [tex]\( x \)[/tex] for which the function is defined.
1. Start by noting that the function [tex]\( f(x) \)[/tex] involves the absolute value expression [tex]\( |2x + 1| \)[/tex].
2. The absolute value function [tex]\( |y| \)[/tex] is defined for all real numbers [tex]\( y \)[/tex].
3. Here, [tex]\( y = 2x + 1 \)[/tex]. Since an absolute value function is always defined regardless of the input, [tex]\( 2x + 1 \)[/tex] can take any real number value.
4. Other operations in the function are:
- Scaling the absolute value by a constant factor [tex]\( -\frac{1}{4} \)[/tex].
- Subtracting 1 from the scaled absolute value.
5. The operations of scaling and subtracting a constant do not place any restrictions on the domain. They are both defined for all real numbers.
Therefore, there are no restrictions on the value of [tex]\( x \)[/tex]. The function [tex]\( f(x) \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
Hence, the domain of the function [tex]\( f(x) = -\frac{1}{4} |2x + 1| - 1 \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
The correct answer is:
D. [tex]\( (-\infty, \infty) \)[/tex]
