Question

Over which interval is this function continually decreasing?

f(x) = |4(x - 1)| + 2
a) (-[infinity], 2)
b) (-[infinity], 1)
c) (2, [infinity])
d) (1, [infinity])

Answer 1

The function f(x) = |4(x - 1)| + 2 is continually decreasing over the interval (1, ∞), corresponding to option (c).

Step-by-step explanation:

The function f(x) = |4(x - 1)| + 2 is a piecewise function with different behaviors for x values less than or greater than 1. Let's consider the two cases:

Case 1: When x < 1

In this case, the function can be simplified as f(x) = -4(x - 1) + 2 = -4x + 6. We can see that as x increases, f(x) decreases, indicating a continuous decrease.

Case 2: When x > 1

In this case, the function can be simplified as f(x) = 4(x - 1) + 2 = 4x - 2. We can see that as x increases, f(x) also increases, indicating a continuous increase.

Therefore, the function f(x) = |4(x - 1)| + 2 is continually decreasing only over the interval (1, ∞), which corresponds to option (c).