Question

Find \((f \circ g)(x)\) and \((g \circ f)(x),\) and graph each of these functions. $$ \begin{array}{l} f(x)=\frac{1}{2} x \\ g(x)=2 \csc x \end{array} $$

Short answer

For \( (f \circ g)(x) = \csc x \). For \( (g \circ f)(x) = 2 \csc (\frac{1}{2} x) \).

Step-by-step solution

Understand the Composition of Functions

To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \). Similarly, to find \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \).

Substitute g(x) into f(x)

For \( (f \circ g)(x) \): \( f(g(x)) = f(2 \csc x) = \frac{1}{2} (2 \csc x) = \csc x \). So, \( (f \circ g)(x) = \csc x \).

Substitute f(x) into g(x)

For \( (g \circ f)(x) \): \( g(f(x)) = g(\frac{1}{2} x) = 2 \csc (\frac{1}{2} x) \). So, \( (g \circ f)(x) = 2 \csc (\frac{1}{2} x) \).

Graph \( (f \circ g)(x) = \csc x \)

The graph of \( \csc x \) will have vertical asymptotes at \( x = n\pi \) (where \( n \) is an integer) and will resemble the reciprocal of the sine function, having peaks and troughs.

Graph \( (g \circ f)(x) = 2 \csc (\frac{1}{2} x) \)

The graph of \( 2 \csc (\frac{1}{2} x) \) will have vertical asymptotes at \( x = 2n\pi \) (where \( n \) is an integer) and will stretch vertically by a factor of 2 compared to \( \csc (\frac{1}{2} x) \), which resembles the reciprocal of the sine function but horizontally compressed by a factor of 2.

Key concepts

Composite Functions

Composite functions involve combining two functions to form a new one. This is done by taking the output of one function and using it as the input for another.
For example, if you have functions \( f(x) \) and \( g(x) \), the composite function \( (f \, \circ \, g)(x) \) means you're plugging \( g(x) \) into \( f(x) \). Similarly, \( (g \, \circ \, f)(x) \) means you're plugging \( f(x) \) into \( g(x) \).

To calculate these, follow the steps:

• Find \( (f \, \circ \, g)(x) \) by substituting \( g(x) \) into \( f(x) \).
  In our exercise, \( f(x) = \frac{1}{2} x \) and \( g(x) = 2 \csc x \). So, \( f(g(x)) = f(2 \csc x) = \frac{1}{2}(2 \csc x) = \csc x \).
• Find \( (g \, \circ \, f)(x) \) by substituting \( f(x) \) into \( g(x) \).
  In our exercise, \( g(f(x)) = g(\frac{1}{2} x) = 2 \csc (\frac{1}{2} x) \).

So, \( (f \, \circ \, g)(x) = \csc x \) and \( (g \, \circ \, f)(x) = 2 \csc (\frac{1}{2} x) \).

Cosecant Function

The cosecant function, written as \( \csc x \), is the reciprocal of the sine function. That is, \( \csc x = \frac{1}{\sin x} \).
However, unlike sine, \( \csc x \) is undefined whenever \( \sin x = 0 \), which happens at integer multiples of \( \pi \) (i.e., \( x = n\pi \) for any integer \( n \)).

Key properties of the cosecant function:

• Vertical asymptotes at \( x = n\pi \), where \( n \) is an integer.
• Unlike sine, which ranges between -1 and 1, \( \csc x \) ranges from \(-\infty\) to -1 and from 1 to \( \infty\).

In our exercise, we noted that \( (f \circ g)(x) \) gave us \( \csc x \). Knowing the properties of \( \csc x \) helps us understand and graph this composite function effectively.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their key features, such as period, amplitude, and asymptotes.

For \( \csc x \), here's what to keep in mind:

• The function has vertical asymptotes where sine is zero (\( x = n\pi \)).
• The graph approaches infinity near these asymptotes.
• It looks like an inverted reciprocal of the sine graph between asymptotes.

For the function \( g \circ f(x) = 2 \csc (\frac{1}{2} x) \), here's what to consider:

• The factor of 2 outside the \( \csc \) function stretches the graph vertically by 2.
• The \( \frac{1}{2} \) inside the \( \csc \) function horizontally stretches the period by a factor of 2 (making the period \( 4\pi \) instead of \( 2\pi \)).
• Vertical asymptotes are now at \( x = 2n \pi \), and the graph still approaches infinity near these points.

Following these guidelines makes it simpler to plot the function accurately.