the graph of (f(x)=|x|) is transformed to (g(x)=|x + 1|-7). on which interval is the function decreasing?\n(…

Explanation:

Step1: Analyze the basic absolute - value function

The parent function $y = |x|$ is decreasing on the interval $(-\infty,0)$ and increasing on the interval $(0,\infty)$.

Step2: Consider the transformation of the function

The function $g(x)=|x + 1|-7$ is a transformation of $y = |x|$. The transformation $y=|x|$ to $y = |x + 1|$ is a horizontal shift to the left by 1 unit, and the subtraction of 7 is a vertical shift down by 7 units. The horizontal shift affects the interval of decrease. For the function $y=|x + 1|$, the vertex is at $x=-1$. Similar to the parent - function $y = |x|$, the function $y=|x + 1|$ (and thus $g(x)=|x + 1|-7$) is decreasing on the interval where $x+1<0$.

Step3: Solve the inequality for the interval of decrease

Solve the inequality $x + 1<0$. Subtract 1 from both sides of the inequality: $x<-1$. So the function $g(x)=|x + 1|-7$ is decreasing on the interval $(-\infty,-1)$.

Answer:

$(-\infty,-1)$