Increasing and Decreasing Functions

Increasing Functions

A function is "increasing" when the y-value increases as the x-value increases, like this:
Increasing Function
It is easy to see that y=f(x) tends to go up as it goes along.

Flat ?

What about that flat bit near the start? Is that OK?
  • Yes, it is OK when we say the function is Increasing
  • But it is not OK if we say the function is Strictly Increasing (no flatness allowed)

Using Algebra

What if we can't plot the graph to see if it is increasing? In that case we need a definition using algebra.

For a function y=f(x):
when x1 < x2 then f(x1) ≤ f(x2) Increasing
when x1 < x2 then f(x1) < f(x2) Strictly Increasing
That has to be true for any x1, x2, not just some nice ones we might choose.
The important parts are the < and ≤ signs ... remember where they go!

An Example:

Increasing Function
This is also an increasing function
even though the rate of increase reduces

For An Interval

Usually we are only interested in some interval, like this one:
Increasing Function
This function is increasing for the interval shown
(it may be increasing or decreasing elsewhere)

Decreasing Functions

The y-value decreases as the x-value increases:
Decreasing Function
For a function y=f(x)
when x1 < x2 then f(x1) ≥ f(x2) Decreasing
when x1 < x2 then f(x1) > f(x2) Strictly Decreasin
:


g
Notice that f(x1) is now larger than (or equal to) f(x2).

An Example

Let us try to find where a function is increasing or decreasing.

Example: f(x) = x3-4x, for x in the interval [-1,2]

Let us plot it, including the interval [-1,2]:
Example Function
Starting from -1 (the beginning of the interval[-1,2]):
  • at x = -1 the function is decreasing,
  • it continues to decrease until about 1.2
  • it then increases from there, past x = 2
Without exact analysis we cannot pinpoint where the curve turns from decreasing to increasing, so let us just say:
Within the interval [-1,2]:
  • the curve decreases in the interval [-1, approx 1.2]
  • the curve increases in the interval [approx 1.2, 2]

Constant Functions

A Constant Function is a horizontal line:
Constant Function

Lines

In fact lines are either increasing, decreasing, or constant.
The equation of a line is:
y = mx + b
The slope m tells us if the function is increasing, decreasing or constant:
m < 0 decreasing
m = 0 constant
m > 0 increasing
 Constant Function

One-to-One

Strictly Increasing (and Strictly Decreasing) functions have a special property called "injective" or "one-to-one" which simply means we never get the same "y" value twice.
General Function Injective Function
General Function "Injective" (one-to-one)
Why is this useful? Because Injective Functions can be reversed!
We can go from a "y" value back to an "x" value (which we can't do when there is more than one possible "x" value).
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