Probably the most important characteristic we want to know about a function is its behavior. That is, where is it increasing and where is it decreasing?

Increasing and decreasing for a function is a comparison between its domain and range, which are both a set of numbers. So, to understand function behavior, we need to first understand number behavior.

Number Behavior: Our real numbers have a natural ordering, which we call less than and greater than. Visually, these mean “to the left of” and “to the right of” on a number line.

If you move to the right on the number line, then the numbers are increasing.

If you move to the left on the number line, then the numbers are decreasing.

With functions, we have two number lines and we compare movement on them to each other.

We want to following this idea through compositions.

Increasing \(\circ \) Increasing

\(f\) and \(g\) are defined graphically below. They are both increasing functions.

What about \(f \circ g\)?

To decide the function behavior of a composition, we need to follow movement through two graphs.

Domain Movement in \(g\): First, we think of movement to the right in the domain of \(f \circ g\). This would be movement to the right in the domain of \(g\). The graph of \(g\) is on the right above. This would be movement to the right on the horizontal axis.

Range Movement in \(g\): Next, according to the graph of \(g\), as we move the right in the domain of \(g\), the values of \(g\) move up, which is to the right in the range of \(g\).

switch: The range of \(g\) becomes the domain of \(f\).

Domain Movement in \(f\): The range of \(g\) becomes the domain of \(f\). We were moving up on the graph of \(g\), which means to the right in the range of \(g\). This switches to movement to the right in the domain of \(f\).

Range Movement in \(f\): Next, according to the graph of \(f\), as we move the right in the domain of \(f\), the values of \(f\) move up, which is to the right in the range of \(f\). This is the range of \(f \circ g\).

Gluing the whole story, movement to the right in the domain of \(f \circ g\) corresponds to movement to the right in the range of \(f \circ g\).

\(f \circ g\) is an increasing function.

We can describe all of this thinking algebraically as well.

If \(a < b\) in the domain of \(f \circ g\), then we want to know about \((f \circ g)(a)\) and \((f \circ g)(b)\).

Step 1: If \(a < b\), then \(g(a) < g(b)\), because \(g\) is an increasing function.

Step 2: Now think of \(g(a)\) and \(g(b)\) as domain numbers for \(f\).

Step 3: If \(g(a) < g(b)\), then \(f(g(a)) < f(g(b))\), because \(f\) is an incresing function.

Beginning to End: If \(a < b\), then \(f(g(a)) < f(g(b))\) and \(f \circ g\) is an increasing function.

Increasing \(\circ \) Decreasing

\(f\) and \(g\) are defined graphically below. \(f\) is an increasing function. \(g\) is a decreasing function.

What about \(f \circ g\)?

To decide the function behavior of a composition, we need to follow movement through two graphs.

Domain Movement in \(g\): First, we think of movement to the right in the domain of \(f \circ g\). This would be movement to the right in the domain of \(g\). The graph of \(g\) is on the right above. This would be movement to the right on the horizontal axis.

Range Movement in \(g\): Next, according to the graph of \(g\), as we move the right in the domain of \(g\), the values of \(g\) move down, which is to the left in the range of \(g\).

switch: The range of \(g\) becomes the domain of \(f\).

Domain Movement in \(f\): The range of \(g\) becomes the domain of \(f\). We were moving down on the graph of \(g\), which means to the left in the range of \(g\). This switches to movement to the left in the domain of \(f\).

Range Movement in \(f\): Next, according to the graph of \(f\), as we move the left in the domain of \(f\), the values of \(f\) move down, which is to the left in the range of \(f\). This is the range of \(f \circ g\).

Gluing the whole story, movement to the right in the domain of \(f \circ g\) corresponds to movement to the left in the range of \(f \circ g\).

\(f \circ g\) is a decreasing function.

We can describe all of this thinking algebraically as well.

If \(a < b\) in the domain of \(f \circ g\), then we want to know about \((f \circ g)(a)\) and \((f \circ g)(b)\).

Step 1: If \(a < b\), then \(g(a) > g(b)\), because \(g\) is an decreasing function.

Step 2: Now think of \(g(a)\) and \(g(b)\) as domain numbers for \(f\).

Step 3: If \(g(a) > g(b)\), then \(f(g(a)) > f(g(b))\), because \(f\) is an increasing function.

Beginning to End: If \(a < b\), then \(f(g(a)) > f(g(b))\) and \(f \circ g\) is a decreasing function.

Decreasing \(\circ \) Increasing

\(f\) and \(g\) are defined graphically below. \(f\) is a decreasing function. \(g\) is an increasing function.

What about \(f \circ g\)?

To decide the function behavior of a composition, we need to follow movement through two graphs.

Domain Movement in \(g\): First, we think of movement to the right in the domain of \(f \circ g\). This would be movement to the right in the domain of \(g\). The graph of \(g\) is on the right above. This would be movement to the right on the horizontal axis.

Range Movement in \(g\): Next, according to the graph of \(g\), as we move the right in the domain of \(g\), the values of \(g\) move up, which is to the right in the range of \(g\).

switch: The range of \(g\) becomes the domain of \(f\).

Domain Movement in \(f\): The range of \(g\) becomes the domain of \(f\). We were moving up on the graph of \(g\), which means to the right in the range of \(g\). This switches to movement to the right in the domain of \(f\).

Range Movement in \(f\): Next, according to the graph of \(f\), as we move the right in the domain of \(f\), the values of \(f\) move down, which is to the left in the range of \(f\). This is the range of \(f \circ g\).

Gluing the whole story, movement to the right in the domain of \(f \circ g\) corresponds to movement to the left in the range of \(f \circ g\).

\(f \circ g\) is a decreasing function.

We can describe all of this thinking algebraically as well.

If \(a < b\) in the domain of \(f \circ g\), then we want to know about \((f \circ g)(a)\) and \((f \circ g)(b)\).

Step 1: If \(a < b\), then \(g(a) < g(b)\), because \(g\) is an increasing function.

Step 2: Now think of \(g(a)\) and \(g(b)\) as domain numbers for \(f\).

Step 3: If \(g(a) < g(b)\), then \(f(g(a)) > f(g(b))\), because \(f\) is a decreasing function.

Beginning to End: If \(a < b\), then \(f(g(a)) > f(g(b))\) and \(f \circ g\) is a decreasing function.

Decreasing \(\circ \) Decreasing

\(f\) and \(g\) are defined graphically below. \(f\) and \(g\) are both decreasing functions.

What about \(f \circ g\)?

To decide the function behavior of a composition, we need to follow movement through two graphs.

Domain Movement in \(g\): First, we think of movement to the right in the domain of \(f \circ g\). This would be movement to the right in the domain of \(g\). The graph of \(g\) is on the right above. This would be movement to the right on the horizontal axis.

Range Movement in \(g\): Next, according to the graph of \(g\), as we move to the right in the domain of \(g\), the values of \(g\) move down, which is to the left in the range of \(g\).

switch: The range of \(g\) becomes the domain of \(f\).

Domain Movement in \(f\): The range of \(g\) becomes the domain of \(f\). We were moving down on the graph of \(g\), which means to the left in the range of \(g\). This switches to movement to the left in the domain of \(f\).

Range Movement in \(f\): Next, according to the graph of \(f\), as we move the left in the domain of \(f\), the values of \(f\) move up, which is to the right in the range of \(f\). This is the range of \(f \circ g\).

Gluing the whole story, movement to the right in the domain of \(f \circ g\) corresponds to movement to the right in the range of \(f \circ g\).

\(f \circ g\) is a increasing function.

Decreasing means turning around and we turned around twice.

We can describe all of this thinking algebraically as well.

If \(a < b\) in the domain of \(f \circ g\), then we want to know about \((f \circ g)(a)\) and \((f \circ g)(b)\).

Step 1: If \(a < b\), then \(g(a) > g(b)\), because \(g\) is a decreasing function.

Step 2: Now think of \(g(a)\) and \(g(b)\) as domain numbers for \(f\).

Step 3: If \(g(a) > g(b)\), then \(f(g(a)) < f(g(b))\), because \(f\) is a decreasing function.

Beginning to End: If \(a < b\), then \(f(g(a)) < f(g(b))\) and \(f \circ g\) is an increasing function.