Question

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

Explanation:

Step1: Recall the parent function

The parent function is \( f(x) = |x| \), which has a vertex at \( (0,0) \) and is decreasing on \( (-\infty, 0) \) and increasing on \( (0, \infty) \).

Step2: Analyze the transformation

For the function \( g(x) = |x + 1| - 7 \), the transformation from \( f(x) = |x| \) involves a horizontal shift left by 1 unit (because of \( x + 1 \)) and a vertical shift down by 7 units (because of \( -7 \)). The vertex of \( g(x) \) is at \( (-1, -7) \).

Step3: Determine the decreasing interval

The absolute - value function \( y = |x - h|+k \) has the same shape as \( y = |x| \) but shifted. For \( g(x)=|x + 1|-7=|x-(-1)|-7 \), the function is decreasing to the left of the vertex \( x=-1 \) and increasing to the right of the vertex \( x = - 1 \). So the function \( g(x) \) is decreasing on the interval \( (-\infty,-1) \).

Answer:

\( (-\infty, - 1) \) (corresponding to the option \( (-\infty, -1) \))