Change

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increasing and decreasing

We use functions to compare information (measurements). One aspect we are interest in is how one measurement changes compared to changes in the other.

When the supply of pineapples goes up does the price go down?
When medicine dosage goes up does the pain go down?
When the speed limit goes up does the number of accidents go up?

Our words for this comparison are increase and decrease.

If the price of pineapples goes down when the supply of pineapples goes up, then we say the price is decreasing with respect to the supply.
If the pain goes down when the dosage goes up, then we say that pain decreases with respect to dosage.
If the number of accidents goes up when the speed limit goes up, then we say that accidents increases with respect to speed limit.

Increase on a Set

The function $f$ increases on the set $S$, if for EVERY pair of numbers $a < b \in S$, we have $f(a) < f(b)$.

When the domain values become greater, than the function values also become greater.

When the domain values become lesser, the function values also become lesser.

The domain and range change in the same way.

Note: You cannot merely test the endpoints of an interval to show a function is increasing. It has to increase for EVERY pair of numbers in the set.

Decrease on a Set

The function $f$ decreases on the set $S$, if for EVERY pair of numbers such that $a < b \in S$, we have $f(a) > f(b)$.

When the domain values become greater, than the function values become lesser.

When the domain values become lesser, the function values become greater.

The domain and range change oppositely.

Note: You cannot merely test the endpoints of an interval to show a function is decreasing. It has to decrease for EVERY pair of numbers in the interval.

Language

In mathematics, the words increase and decrease are seemingly used in two different ways.

$\blacktriangleright $ A List of Numbers: The domain and range of a function are essentially a list of numbers. When we present a list of numbers, we line them up one after the other and that gives them a natural ordering. Each number sits at a position before or after other numbers.

As we step through the positions, the values of the numbers become greater, or become lesser, or stay the same.

If the value of the numbers become greater as we step through the positions of the numbers, then we say the list (sequence) of numbers increases.
If the value of the numbers become lesser as we step through the positions of the numbers, then we say the list (sequence) of numbers decreases.

For lists of numbers (sequences), we use the words increase and decrease as a comparison between the change in the values of the numbers compared to the change in their position in the list.

We can extend this idea to a comparison of function values and domain numbers, essentially two lists.

$\blacktriangleright $ A Function:

A function is also kind of like a list of numbers with positions. The domain numbers play the role of positions and the function values play the role of list numbers sitting in those positions.

We compare the change in the function numbers to the change in the domain numbers.

If the function values become greater as the domain numbers become greater, then we say the function increases.
If the function values become lesser as the domain numbers become greater, then we say the function decreases.

We have a tendency to read left to right with graphs, but that is only half the story. Everything can be read right to left.

If the function values become greater as the domain numbers become lesser, then we say the function decreases.
If the function values become lesser as the domain numbers become lesser, then we say the function increases.

This tells us that increasing and decreasing are two-way comparisons.

If the function values and domain numbers change in the same way (both become greater or lesser), then we say the function increases.
If the function values and domain numbers change in opposite ways (one greater and one lesser), then we say the function decreases.

Graphically, increasing appears as dots to the right being higher than dots to the left. Decreasing appears as dots on the curve being lower to the right.

Or, the reverse....

Increasing also appears as dots to the left being lower than dots to the right.

Decreasing also appears as dots on the curve being higher to the left.

Increasing and Decreasing

Let $g(k)$ be a function. The graph of $y= g(k)$ is displayed below.

$g$ is increasing decreasing neither on the interval $(-8,-3)$.
$g$ is increasing decreasing neither on the interval $[-1,3)$.
$g$ is increasing decreasing neither on the interval $[4, 8)$.

In the example above, $g$ is decreasing on $(-8,-3)$. Is $g$ decreasing on any and every subinterval of $(-8,-3)$.

Yes No

Decreasing

Let $m(t)$ be a function. The graph of $y = m(t)$ is displayed below.

$m$ is increasing decreasing neither on the interval $(-8,-3)$.
$m$ is increasing decreasing neither on the interval $[-3, 8)$.

However, $m$ is not decreasing on the interval $(-8, 8)$. To show that a function is not decreasing on an interval, we merely have to give ONE counterexample. That is we need to come up with ONE pair of numbers $a < b$ in the interval where $f(a) \ngtr f(b)$. Let’s select $a = -4$ and $b = -3$.

\[ -4 < -3 \text { but, } f(-4) = 4 \ngtr f(-3) = 6 \]

Counterexample

A counterexample is one example where all of the conditions of a statement are met, yet the conclusion is not true.

One single counterexample shows a statement to be false.

Decreasing

Let $N(z)$ be a function. The graph of $y = N(z)$ is displayed below.

Select all of the correct statements

$N$ is decreasing on the interval $(-3,2)$ $N$ is decreasing on the interval $(2,8)$ $N$ is decreasing on the interval $(-3,8)$ $N$ is decreasing on the interval $[-3,8)$ $N$ is decreasing on the interval $(-3,8]$ $N$ is decreasing on the interval $[-3,8]$

Measuring Rates

Comparing how the connected values in the domain and range change is called a rate. Our symbol for a change in an amount is a Greek uppercase delta, $\Delta $. And, we use fractions to quantify the comparison of changes.

\[ rate = \frac {\Delta \, pineapples}{\Delta \, price}\]

If the changes in pineapples and price are both positive, then this rate is positive.
If the changes in pineapples and price are both negative, then this rate is again positive.
If the changes in pineapples and price are of different sign, one increases while the other decreases, then this rate is negative.

Using rates, we can compare and measure changes in function values over a domain interval.

Rate of Change

The rate-of-change of a function $f$ across $[a, b]$ in the domain is given by

\[ \frac {f(b) - f(a)}{b - a} \]

This is also referred to as the average rate of change across the interval.

The rate-of-change measures how the function values change relative to changes in the domain.

Rates

Let $V(x)$ be a function. The graph of $y = V(x)$ is displayed below.

From the graph, we can estimate that over the interval $[-7, -5]$, $V$ had a rate-of-change of $\frac {-6 - 0}{-5 - (-7)} = \frac {-6}{2} = -3$.

From the graph, we can estimate that over the interval $[1, 8]$, $V$ had a rate-of-change of $\frac {-8 - (-1)}{8 - 1} = \frac {-7}{7} = -1$.

All of this might sound familiar. You have probably seen this when working with slopes of lines, $\frac {rise}{run}$. The rise is the vertical change (change in the value of the function). The run is the horizontal change (change in the domain).

$\blacktriangleright $ Rate-of-change is a function idea that we will connect up to the geometric idea of slope.

The rate of change over an interval is an overall measurement. It compares the changes occuring over the whole interval. It compares final values to initial values.

Let $M(t)$ be a function. The graph of $y = M(t)$ is displayed below.

The rate of change of $M$ over the interval $[-5,5]$ is

positive negative

The rate of change of $M$ over the interval $[-2,1]$ is

positive negative

If the rate of change of a function is positive over the interval $[a,b]$ then the rate of change over any subinterval must also be positive.

True False

Constant Rate of Change

Linear functions are functions that can be represented with formulas of the form $L(x) = A \, x + B$.

Given an interval, we can measure the change of a linear function over this interval.

Let $L(x)$ be the linear function $L(x) = \frac {1}{2} x - 4$. The graph of $y = L(x)$ is displayed below.

The rate of change of $L$ over the interval $[-6, 0]$ is

\[ \frac {L(0) - L(-6)}{0 - (-6)} = \frac {-4 - (-7)}{6} = \frac {3}{6} = \frac {1}{2} \]

The rate of change of $L$ over the interval $[-4, 6]$ is

\[ \frac {L(6) - L(-4)}{6 - (-4)} = \frac {-1 - (-6)}{10} = \frac {5}{10} = \frac {1}{2} \]

The rate of change of $L$ over the interval $[-2, 2]$ is

\[ \frac {L(2) - L(-2)}{2 - (-2)} = \frac {-3 - (-5)}{4} = \frac {2}{4} = \frac {1}{2} \]

The rate of change of $L$ over the interval $[8, 10]$ is

\[ \frac {L(10) - L(8)}{10 - 8} = \frac {1 - 0}{2} = \frac {1}{2} \]

Linear functions have a special property that other functions do not have. No matter what interval you select, the rate of change over that interval is always the same.

Let $L$ be the linear function in the example above, $L(x) = \frac {1}{2} x - 4$.

Let $a$ and $b$ be any distinct real numbers.

The rate of change of $L$ over the interval $[a, b]$ is

\[ \frac {L(b) - L(a)}{b - a} = \frac {\left ( \frac {b}{2} - 4 \right ) - \left ( \frac {a}{2} - 4 \right )}{b-a} = \frac {\left ( \frac {b - a}{2} \right )}{b-a} = \frac {1}{2} \]

$\blacktriangleright $ Linear functions have a constant rate of change, which measures the constant slope of the corresponding line.

We will use this fact to help us understand how function values change for any function.

A Peek Ahead...

Later in Calculus, we will more closely examine sequences. We will describe and represent them as functions. Their domains will be the natural numbers, rather than the real numbers.

That will bring all of our versions of increasing and decreasing together under one umbrella.

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2026-05-24 00:17:48