Question

select the correct answer.
what is the domain of this function?
$f(x) = \frac{3}{4} | x - 3 | + 1$
a. $1, \infty)$
b. $(-\infty, \infty)$
c. $(-\infty, 3)$
d. $(-\infty, \frac{3}{4}$

Explanation:

Step1: Recall the domain of absolute value function

The absolute value function \(|x|\) is defined for all real numbers \(x\). In the given function \(f(x)=\frac{3}{4}|x - 3|+1\), the expression inside the absolute value is \(x - 3\), which is a linear expression. Linear expressions are defined for all real numbers.

Step2: Determine the domain of \(f(x)\)

Since there are no restrictions (like division by zero or square root of a negative number) in the function \(f(x)=\frac{3}{4}|x - 3|+1\), the input \(x\) can be any real number. So the domain is all real numbers, which is represented as \((-\infty,\infty)\).

Answer:

B. \((-\infty, \infty)\)