Question

what is the range of this function?
y = |-4x| - 4
{y|y > -4}
{y|y ≥ 0}
{y|y ≥ -4}
all real numbers

Explanation:

Step1: Analyze the absolute value

The absolute value of any real number is non - negative. For the expression \(|-4x|\), we know that \(|-4x|\geq0\) for all real numbers \(x\). This is because the absolute value function \(|a|\) is defined as \(a\) when \(a\geq0\) and \(-a\) when \(a < 0\), and in either case, the result is greater than or equal to \(0\).

Step2: Analyze the function \(y = |-4x|-4\)

We start with \(|-4x|\geq0\). If we subtract \(4\) from both sides of the inequality \(|-4x|\geq0\), we get \(|-4x|-4\geq0 - 4\), which simplifies to \(y=|-4x|-4\geq - 4\). So the range of the function \(y = |-4x|-4\) is all real numbers \(y\) such that \(y\geq - 4\), or in set - builder notation \(\{y|y\geq - 4\}\).

Answer:

\(\{y|y\geq - 4\}\)