Question

Find \((f \circ g)(x)\) and \((g \circ f)(x),\) and graph each of these functions. $$ \begin{array}{l} f(x)=-2 x \\ g(x)=\cot x \end{array} $$

Short answer

(f \[\circ\] g)(x) = -2 \cot x, (g \[\circ\] f)(x) = \cot(-2x). Graphs include transformations of \cot x.

Step-by-step solution

Understand the composition of functions

To find the composition of two functions, first understand that \( (f \circ g)(x)=(f(g(x))) \). This means that we substitute the function \(g(x) \) into the function \(f(x) \). Similarly, \( (g \circ f)(x)=(g(f(x))) \) involves substituting \(f(x) \) into \(g(x) \).

Find \( (f \circ g)(x) \)

Given, \( g(x)=\cot x \), substitute this into \( f(x)=-2x \): \ f(g(x)) = f(\cot x) = -2(\cot x) = -2 \cot x \.

Find \( (g \circ f)(x) \)

Given, \( f(x)=-2x \), substitute this into \( g(x)=\cot x \): \ g(f(x)) = g(-2x) = \cot(-2x) \.

Graph each function

To graph \( (f \circ g)(x) = -2 \cot x \), plot the standard \cot x \ function and apply the transformation by multiplying by -2, which mirrors and stretches the graph. For \( (g \circ f)(x) = \cot(-2x) \), plot \cot x \ applying the horizontal compression and reflection across the y-axis due to the -2 coefficient.

Key concepts

Composite Functions

Composite functions involve the combination of two functions to create a new function. Mathematically, if we have two functions, say \(f(x)\) and \(g(x)\), then the composite functions \((f \,\circ\, g)(x)\) and \((g \,\circ\, f)(x)\) can be found.

Understanding this concept is critical. When you see \((f \,\circ\, g)(x)\), it means you need to substitute the whole function \(g(x)\) into the function \(f(x)\). Similarly, \((g \,\circ\, f)(x)\) requires you to put the function \(f(x)\) inside the function \(g(x)\).

In our specific case:

• \(f(x) = -2x\)
• \(g(x) = \cot x\)

To find \((f \,\circ\, g)(x)\), substitute \(g(x)\) into \(f(x)\):
\(f(g(x)) = f(\cot x) = -2(\cot x) = -2 \cot x\)

To find \((g \,\circ\, f)(x)\), substitute \(f(x)\) into \(g(x)\):
\(g(f(x)) = g(-2x) = \cot(-2x)\)
This gives us our composite functions:

• \((f \,\circ\, g)(x) = -2 \cot x\)
• \((g \,\circ\, f)(x) = \cot(-2x)\)

Trigonometric Functions

Trigonometric functions play an essential role in many areas of mathematics and science. In our problem, the trigonometric function used is \(\cot x\).

The cotangent function, \(\cot x\), is the reciprocal of the tangent function. Thus, \(\cot x = \frac{1}{\tan x}\). It has a period of \(\pi\) and is undefined where \(\tan x = 0\), i.e., at points where \( x = n\pi \) for integer \(n\).

Here are some key properties of \(\cot x\):

• \(\cot(x)\) has a period of \(\pi\).
• It is an odd function, meaning \(\cot(-x) = -\cot(x)\).
• The function approaches infinity as it nears the undefined points (asymptotes at multiples of \(\pi\)).

In the solution, we utilize \(\cot x\) to create composite functions. When we compose functions using a trigonometric function, it's important to remember the properties and behavior of the original function.

For instance:

• \(\cot(x)\) and \(-2 \cot(x)\) stretch and reflect the graph but maintain the same asymptotic behavior.
• \(\cot(-2x)\) applies transformations that compress and reflect the graph horizontally.

Graph Transformations

Graph transformations help us visualize how changes to the function impact its shape and position on a graph.

For the functions \((f \circ g)(x) = -2 \cot x\) and \((g \circ f)(x) = \cot(-2x)\), several transformations occur:

### Transformations of \( -2 \cot x\):

• \(\cot x\) is reflected across the x-axis (due to the negative sign).
• The graph is vertically stretched by a factor of 2 (because of the multiplier -2).

### Transformations of \( \cot(-2x)\):

• \(\cot x\) undergoes a horizontal compression by a factor of 2 (due to the coefficient of -2).
• The graph is reflected across the y-axis (because of the negative sign within the function).

When graphing these transformed functions:

• Identify key points and asymptotes on the standard cotangent graph.
• Apply the respective transformations—vertical stretch, horizontal compression, and reflections.
• Ensure that the asymptotic behaviors align with the transformed functions.

Both transformations are essential to fully understand the resulting functions and accurately graph them.
This deepens our insights into how composite functions involving trigonometric components behave on a graph.