the graph of $f(x)=|x|$ is transformed to $g(x)=|x + 1|-7$. on which interval is the function…
the graph of $f(x)=|x|$ is transformed to $g(x)=|x + 1|-7$. on which interval is the function decreasing?\n$(-\\infty,-7)$\n$(-\\infty,-1)$\n$(-\\infty,1)$\n$(-\\infty,7)$
Answer
Explanation:
Step1: Analyze the basic absolute - value function
The basic absolute - value 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 (y = |x|), the interval of decrease is (x<0). For (y=|x + 1|), we set (x+1<0).
Step3: Solve the inequality for the interval of decrease
Solve (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))