Identify intervals on which the function is increasing, decreasing, or constant.
-$$f ( x ) = | x - 7 | - 3$$
A) Increasing: $( - 7 , \infty )$; decreasing: $( - \infty , - 7 )$
B) Increasing: $( - 3 , \infty )$; decreasing: $( - \infty , - 3 )$
C) Increasing: $( 7 , \infty )$; decreasing: $( - \infty , 7 )$
D) increasing: $( - \infty , 7 )$; decreasing: $( 7 , \infty )$

Question

Quizplus · Accepted Answer

The function $f(x) = |x - 7| - 3$ is increasing for $x > 7$ because as $x$ increases beyond 7, the value of $|x - 7|$ increases, making $f(x)$ increase. It is decreasing for $x < 7$ because as $x$ decreases from 7, the value of $|x - 7|$ increases (since we are moving further away from 7 in the negative direction), but since the absolute value reflects this as a positive increase, $f(x)$ actually decreases in value. There is no interval on which the function is constant.

Identify Intervals on Which the Function Is Increasing, Decreasing, or Constant