1.7. Increasing and Decreasing http://www.ck12.org
Solution:The green function seems to be discrete values along the liney=x+2. While the discrete values clearly
increase and the line would be monotonically increasing, these values are missing a key part of what it means to be
monotonic. The green function does not have a positive slope and is therefore not monotonic.
The red function also seems to be increasing, but the slope at everyxvalue is zero. In Calculus the definition of
monotonic will be refined to handle special cases like this. For now, this function is not monotonic.
The blue function seems to bey=√x−2, is increasing everywhere that is visible, and probably extends to the right.
This function is monotonic where the function is defined forx∈( 0 ,∞).
Example C
Estimate the intervals where the function is increasing and decreasing.
Solution:
Increasing:x∈(−∞,− 4 )∪(− 2 , 1. 5 )
Decreasing:x∈(− 4 ,− 2 )∪( 1. 5 ,∞)
Note that open intervals are used because atx=− 4 ,− 2 , 1 .5 the slope of the function is zero. This is where the slope
transitions from being positive to negative. The reason why open parentheses are used is because the function is not
actually increasing or decreasing at those specific points.
Concept Problem Revisited
Increasing is where the function has a positive slope and decreasing is where the function has a negative slope. A
common misconception is to look at the squaring function and see two curves that symmetrically increase away
from zero. Instead, you should always read functions from left to right and draw slope lines and decide if they are
positive or negative.