Rate for Composition

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rates of rates

Rates of Change is our measurement of function behavior.

$\vartriangleright $ Functions increase. However, they could increase slowly with a small positive rate of change. They could increase quickly with a large positive rate of change.

$\vartriangleright $ Functions decrease. However, they could decrease slowly with a small negative rate of change. They could decrease quickly with a large negative rate of change.

What about compositions?

What happens when you

compose an increasing function with an increasing function?
compose an increasing function with an decreasing function?
compose an decreasing function with an increasing function?
compose an decreasing function with an decreasing function?

To investigate these we need to remember the definitions of increasing and decreasing.

Increasing

Let $f$ be a function defined on the domain $D$.
Let $S \subset D$ be any subset of $D$.

$f$ is increasing on $S$ provided $f$ possesses this property:

For every pair $a, b \in S$, when $a \leq b$ then $f(a) \leq f(b)$.

Decreasing

Let $f$ be a function defined on the domain $D$.
Let $S \subset D$ be any subset of $D$.

$f$ is decreasing on $S$ provided $f$ possesses this property:

For every pair $a, b \in S$, when $a \leq b$ then $f(a) \geq f(b)$.

Increasing $\circ $ Increasing

Suppose that both $F$ and $G$ are increasing functions.

$\vartriangleright $ That means that when the numbers going into $F$ are increasing, then the numbers coming out of $F$ are increasing.

$\vartriangleright $ That means that when the numbers going into $G$ are increasing, then the numbers coming out of $G$ are increasing.

Now, consider the composition, $F \circ G$.

Suppose the numbers going in $F \circ G$ are increasing. What are the output numbers doing?

\[ (F \circ G)(x) = F(G(x)) \]

Pretend that the values of $x$ are increasing. Then the values of $G(x)$ are increasing, since $G$ is an increasing function.

These values of $G(x)$, which are increasing, are going into $F$. Therefore, the output of $F$ is increasing. But, these are the values of the composition.

When $x$ increases, then $(F \circ G)(x) = F(G(x))$ increases.

$increasing \circ increasing = increasing$

Let $F$ be an increasing function.
Let $G$ be an increasing function.

Consider, $F \circ G$.

Suppose $a$ and $b$ are in the domain of $F \circ G$, with $a < b$.

Then $G(a) < G(b)$, because $G$ is an increasing function.

Then $F(G(a)) < F(G(b))$, because $F$ is an increasing function.

Increasing $\circ $ Decreasing

Suppose that $F$ is an increasing function.
Suppose that $G$ is a decreasing function.

$\vartriangleright $ That means that when the numbers going into $F$ are increasing, then the numbers coming out of $F$ are increasing.

$\vartriangleright $ It also means that when the numbers going into $F$ are decreasing, then the numbers coming out of $F$ are decreasing.

$\vartriangleright $ That means that when the numbers going into $G$ are increasing, then the numbers coming out of $G$ are decreasing.

Now, consider the composition, $F \circ G$.

Suppose the numbers going in $F \circ G$ are increasing. What are the output numbers doing?

\[ (F \circ G)(x) = F(G(x)) \]

Pretend that the values of $x$ are increasing. Then the values of $G(x)$ are decreasing, since $G$ is a decreasing function.

These values of $G(x)$, which are decreasing, are going into $F$. Therefore, the output of $F$ is decreasing, since $F$ is an increasing function. But, these are the values of the composition.

When $x$ increases, then $(F \circ G)(x) = F(G(x))$ decreases.

$increasing \circ decreasing = decreasing$

Let $F$ be an increasing function.
Let $G$ be a decreasing function.

Consider, $F \circ G$.

Suppose $a$ and $b$ are in the domain of $F \circ G$, with $a < b$.

Then $G(b) < G(a)$, because $G$ is a decreasing function.

Then $F(G(b)) < F(G(a))$, because $F$ is an increasing function.

Decreasing $\circ $ Increasing

Suppose that $F$ is a decreasing function.
Suppose that $G$ is an increasing function.

$\vartriangleright $ That means that when the numbers going into $F$ are increasing, then the numbers coming out of $F$ are decreasing.

$\vartriangleright $ It also means that when the numbers going into $F$ are decreasing, then the numbers coming out of $F$ are increasing.

$\vartriangleright $ That means that when the numbers going into $G$ are increasing, then the numbers coming out of $G$ are increasing.

$\vartriangleright $ It also means that when the numbers going into $G$ are decreasing, then the numbers coming out of $G$ are decreasing.

Now, consider the composition, $F \circ G$.

Suppose the numbers going in $F \circ G$ are increasing. What are the output numbers doing?

\[ (F \circ G)(x) = F(G(x)) \]

Pretend that the values of $x$ are increasing. Then the values of $G(x)$ are increasing, since $G$ is a increasing function.

These values of $G(x)$, which are increasing, are going into $F$. Therefore, the output of $F$ is decreasing, since $F$ is an decreasing function. But, these are the values of the composition.

When $x$ increases, then $(F \circ G)(x) = F(G(x))$ decreases.

$decreasing \circ increasing = decreasing$

Let $F$ be a decreasing function.
Let $G$ be an increasing function.

Consider, $F \circ G$.

Suppose $a$ and $b$ are in the domain of $F \circ G$, with $a < b$.

Then $G(a) < G(b)$, because $G$ is an increasing function.

Then $F(G(b)) < F(G(a))$, because $F$ is an decreasing function.

Decreasing $\circ $ Decreasing

Suppose that both $F$ and $G$ are decreasing functions.

$\vartriangleright $ That means that when the numbers going into $F$ are increasing, then the numbers coming out of $F$ are decreasing.

$\vartriangleright $ It also means that when the numbers going into $F$ are decreasing, then the numbers coming out of $F$ are increasing.

$\vartriangleright $ That means that when the numbers going into $G$ are increasing, then the numbers coming out of $G$ are decreasing.

$\vartriangleright $ It also means that when the numbers going into $G$ are decreasing, then the numbers coming out of $G$ are increasing.

Now, consider the composition, $F \circ G$.

Suppose the numbers going in $F \circ G$ are increasing. What are the output numbers doing?

\[ (F \circ G)(x) = F(G(x)) \]

Pretend that the values of $x$ are increasing. Then the values of $G(x)$ are decreasing, since $G$ is an decreasing function.

These values of $G(x)$, which are decreasing, are going into $F$. Therefore, the output of $F$ is increasing, because $F$ is a decreasing function. But, these are the values of the composition.

When $x$ increases, then $(F \circ G)(x) = F(G(x))$ increases.

$decreasing \circ decreasing = increasing$

Let $F$ be a decreasing function.
Let $G$ be a decreasing function.

Consider, $F \circ G$.

Suppose $a$ and $b$ are in the domain of $F \circ G$, with $a < b$.

Then $G(a) > G(b)$, because $G$ is a decreasing function.

Then $F(G(a)) < F(G(b))$, because $F$ is a decreasing function.

$e^{-(x+3)^2}$

$ln(x^2 + 3x + 5)$

Quartic

Let $F(x) = (x+7)(x-1)$

Let $G(k) = (k+4)(k-3)$

Where is the composition, $F \circ G$, increasing and decreasing?

$\vartriangleright $ $(F \circ G)(t) = t^4 +2 t^3 -17 t^2 - 18 t + 65$

We can approximate the critical numbers from the graph.

This browser does not support embedded elements.

The critical numbers are approximately $-3.541$, $-0.5$, and $2.541$.

Algebraically

Now, let’s obtain the critical numbers algebraically.

We have a composition of two quadratic functions. We need to know where they individually increase and decrease.

Both $F$ and $G$ have a positive leading coefficient. Therefore, they both decrease and the increase. The derivative will reveal the specifics.

$\blacktriangleright $ $F'(x) = 2x + 6$

This tells us that the critical number is $-3$. $F$ decreases on $(-\infty , -3)$ and increases on $(-3, \infty )$.

$\blacktriangleright $ $G'(k) = 2k + 1$

This tells us that the critical number is $-\frac {1}{2}$. $F$ decreases on $(-\infty , -\frac {1}{2})$ and increases on $(-\frac {1}{2}, \infty )$.

$\blacktriangleright $ Overlaping Intervals

We need to know when the output from $G$ crosses $-3$, which is where $F$ changes its behavior.

\begin{align*} (k+4)(k-3) & = -3 \\ k^2 + k - 12 & = -3 \\ k^2 + k - 9 & = 0 \end{align*}

\[ k = \frac {-1 \pm \sqrt {1^2 - 4 (1) (-9)}}{2} = \frac {-1 \pm \sqrt {37}}{2} \]

We are feeling good, because $\frac {-1 - \sqrt {37}}{2} \approx -3.541$ and $\frac {-1 + \sqrt {37}}{2} \approx 2.541$, which are the approximations from the graph.

$\blacktriangleright (-\infty , \frac {-1 - \sqrt {37}}{2})$

On this interval $G$ is a decreasing function.
The range of $G$ on this interval is $(-3, \infty )$.
On $(-3, \infty )$, $F$ is an increasing function.
Therefore, the composition $F \circ G$ is a decreasing function.

$\blacktriangleright (\frac {-1 - \sqrt {37}}{2}, \frac {1}{2})$

On this interval $G$ is a decreasing function.
On this interval, $G < -3$.
On $(-\infty , -3)$, $F$ is a decreasing function.
Therefore, the composition $F \circ G$ is an increasing function.

$\blacktriangleright (\frac {1}{2}, \frac {-1 + \sqrt {37}}{2})$

On this interval $G$ is an increasing function.
On this interval, $G < -3$.
On $(-\infty , -3)$, $F$ is a decreasing function.
Therefore, the composition $F \circ G$ is a decreasing function.

$\blacktriangleright (\frac {-1 + \sqrt {37}}{2}, \infty )$

On this interval $G$ is an increasing function.
On this interval, $G > -3$.
On $(-3, \infty )$, $F$ is an increasing function.
Therefore, the composition $F \circ G$ is an increasing function.

We now know that

$F \circ G$ decreases on $(-\infty , \frac {-1 - \sqrt {37}}{2})$
$F \circ G$ increases on $(\frac {-1 - \sqrt {37}}{2}, \frac {1}{2})$
$F \circ G$ decreases on $(\frac {1}{2}, \frac {-1 + \sqrt {37}}{2})$
$F \circ G$ increases on $(\frac {-1 + \sqrt {37}}{2}, \infty )$

At $\frac {-1 - \sqrt {37}}{2}$ there is a local minimum of $(F \circ G)\left ( \frac {-1 - \sqrt {37}}{2} \right )$

At $-\frac {1}{2}$ there is a local maximum of $(F \circ G)\left ( -\frac {1}{2} \right )$

At $\frac {-1 + \sqrt {37}}{2}$ there is a local minimum of $(F \circ G)\left ( \frac {-1 + \sqrt {37}}{2} \right )$

How do these compare with our graph information?

\[ (F \circ G)\left ( \frac {-1 - \sqrt {37}}{2} \right ) = -16 \]

That is what the graph says!

\[ (F \circ G)\left ( -\frac {1}{2} \right ) = \frac {1113}{16} \approx 69.5625 \]

That is what the graph says!

Wonderful !!!

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more examples can be found by following this link
More Examples of Composition

2025-01-07 00:25:59

Increasing \(\circ \) Increasing

Increasing \(\circ \) Decreasing

Decreasing \(\circ \) Increasing

Decreasing \(\circ \) Decreasing