The graph of [tex]f(x)=|x|[/tex] is transformed to [tex]g(x)=|x+1|-7[/tex]. On which interval is the function decreasing?
A. [tex](-\infty,-7)[/tex]
B. [tex](-\infty,-1)[/tex]
C. [tex](-\infty, 1)[/tex]
D. [tex](-\infty, 7)[/tex]
Answer (1)
The function g ( x ) = ∣ x + 1∣ − 7 is a transformation of the absolute value function.
The vertex of the absolute value function is shifted to x = − 1 .
The function is decreasing for x < − 1 .
The interval on which the function is decreasing is ( − ∞ , − 1 ) .
Explanation
Understanding the Function The given function is g ( x ) = ∣ x + 1∣ − 7 . We need to find the interval where this function is decreasing.
Analyzing the Transformation The absolute value function f ( x ) = ∣ x ∣ has a vertex at x = 0 . The transformation g ( x ) = ∣ x + 1∣ − 7 shifts the vertex to x = − 1 and shifts the entire graph down by 7 units.
Determining the Decreasing Interval The absolute value function ∣ x + 1∣ is decreasing for x < − 1 and increasing for -1"> x > − 1 . The vertical shift by − 7 does not affect the intervals where the function is increasing or decreasing.
Conclusion Therefore, the function g ( x ) = ∣ x + 1∣ − 7 is decreasing on the interval ( − ∞ , − 1 ) .
Examples
Imagine you're walking along a V-shaped path. The lowest point of the 'V' is at x = − 1 . As you walk from the left towards x = − 1 , you're going downhill (decreasing). Once you pass x = − 1 , you start walking uphill (increasing). This problem helps understand how transformations affect the direction of a function, which is useful in various real-world scenarios like optimizing costs or modeling physical phenomena.
