To determine the domain of the function [tex]\( f(x) = \frac{3}{4} |x - 3| + 1 \)[/tex], we need to consider the definition and components of the function.
1. Absolute Value Function: The absolute value function [tex]\( |x - 3| \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This is because taking the absolute value of any real number [tex]\( x - 3 \)[/tex] always yields a non-negative real number. Therefore, there are no restrictions on [tex]\( x \)[/tex] from the absolute value component.
2. Linear Operations: The terms involving multiplication by [tex]\(\frac{3}{4}\)[/tex] and addition by [tex]\(1\)[/tex] are linear operations. Multiplying by a constant [tex]\(\frac{3}{4}\)[/tex] and then adding another constant [tex]\(1\)[/tex] are operations that do not impose any restrictions on [tex]\( x \)[/tex].
Since neither the absolute value operation nor the subsequent linear transformations impose any restrictions on [tex]\( x \)[/tex], the function [tex]\( f(x) \)[/tex] is defined for all real numbers.
Thus, the domain of the function [tex]\( f(x) = \frac{3}{4} |x - 3| + 1 \)[/tex] is all real numbers. This is expressed in interval notation as:
[tex]\[ (-\infty, \infty) \][/tex]
Based on the analysis, the correct answer is:
C. [tex]\( (-\infty, \infty) \)[/tex]
