Question

what is the domain of the function $f(x) = 3|x + 4| + 1$?
● all real numbers
● all real numbers less than or equal to $-4$
● all real numbers greater than or equal to $1$
● all real numbers greater than or equal to $-4$

Explanation:

Step1: Recall domain definition

The domain of a function is the set of all possible input values (x - values) for which the function is defined.

Step2: Analyze the function type

The function \( f(x)=3|x + 4|+1 \) is an absolute - value function. The absolute - value function \( y = |u|\) is defined for all real numbers \( u \). In the function \( f(x)\), the expression inside the absolute - value is \( u=x + 4 \), and \( x+4 \) is a linear expression (a polynomial of degree 1), which is defined for all real numbers \( x \). There are no restrictions such as division by zero (since there is no denominator) or taking the square root of a negative number (since there is no square root operation) in this function. So, the function \( f(x)=3|x + 4|+1 \) is defined for all real numbers \( x \).

Step3: Verify with the graph

Looking at the graph of the absolute - value function (a V - shaped graph), we can see that the graph extends infinitely to the left and to the right along the x - axis. This means that for every real number \( x \) (positive, negative, and zero), there is a corresponding point on the graph, which implies that the domain includes all real numbers.

Answer:

all real numbers