Consider the function, \(y = d(x)\), shown here.

The graph of this function models damped sinusoidal motion, typical of a free-swinging pendulum.

Identify two component functions, \(f\) and \(g\), whose product or quotient function would model behaviour similar to \(y = d(x)\)? Provide reasoning for your choice of functions. Use graphing technology to verify your choice.

Solution

One of the component functions is sinusoidal.

Since a local maximum occurs at \(x = 0\), it is easiest to use the cosine function.

Recall, for \(y = a\cos(x)\), the amplitude is \(\lvert a \rvert\).

We seek a product function, \(y = f(x)\cos(x)\), such that the function \(f(x)\) causes a “varying amplitude” for the curve, so the local maximum and minimum values will be determined by the value of \(f(x)\) at specific values of \(x\).

For \(y = d(x)\), the height of each successive peak gradually decreases, approaching \(0\) as \(x \rightarrow \infty\).

Thus, \(f(x)\) must be a decreasing function that approaches a horizontal asymptote of \(y = 0\) as \(x\) increases in value.

Also, note that \(d(x)\) has neither even nor odd symmetry.

The function \(y = \cos(x)\) is even so \(f(x)\) must be neither even nor odd.

Identify two component functions, \(f\) and \(g\), whose product or quotient function would model behaviour similar to \(y = d(x)\)? Provide reasoning for your choice of functions. Use graphing technology to verify your choice.

Solution

Of the various functions we have studied in this course, we know that \(y = a^x, a \neq 0\), is neither an even nor an odd function, and it is an increasing function over its domain when \(a \gt 1\) and a decreasing function when \(0 \lt a \lt 1\).

For example, \(y = \left(\dfrac{1}{2}\right)^{x}\) is a decreasing function that approaches a horizontal asymptote of \(y = 0\) as \(x \rightarrow \infty\).

However, this function decreases fairly quickly.

A more gradually decreasing exponential function will have a base slightly less than \(1\), such as \(y = (0.9)^{x}\).

Identify two component functions, \(f\) and \(g\), whose product or quotient function would model behaviour similar to \(y = d(x)\)? Provide reasoning for your choice of functions. Use graphing technology to verify your choice.

Solution

Does \(y = (0.9)^{x}\cos(x)\) model behaviour similar to \(y = d(x)\)?

We can obtain the graph of \(y = (0.9)^{x}\cos(x)\) using graphing technology.

This function models behaviour similar to that of \(y = d(x)\).

The rate at which the sinusoidal function is dampened can be accomplished by varying the base of the exponential function.

Identify two component functions, \(f\) and \(g\), whose product or quotient function would model behaviour similar to \(y = d(x)\)? Provide reasoning for your choice of functions. Use graphing technology to verify your choice.

Solution

We should also note that this behaviour can be obtained by the quotient of a sinusoidal function and an exponential function, \(y = a^{x}\), where \(a \gt 1\).

For example, the function \(y = \dfrac{\cos(x)}{(1.1)^{x}}\) will model behaviour similar to \(y = d(x)\).

This curve can also be defined by the product function \(y = (1.1)^{-x}\cos(x)\).

The most common form of sinusoidal damping is exponential damping, where the amplitude of successive peaks undergoes exponential decay.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/vf8tn87t#

Product and quotients of functions can be used to model many different, complicated situations.

Understanding the behaviour of the component functions helps in determining the properties and sketching the graphs of these, more complex, combined functions.

Two functions can be combined using multiplication or division to create a new (and often more complex) function.

• The product of two functions \(f\) and \(g\) is a function defined by \((f \cdot g)(x) = f(x) \cdot g(x)\).
• The quotient of two functions \(f\) and \(g\) is a function defined by \(\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}\) where \(g(x) \neq 0\).
• The domain of the product or quotient function is the set of all real numbers in the domain of both \(f\) and \(g\), with an added restriction of \(g(x) \neq 0\) for \(\dfrac{f}{g}\).
• The graph of \(f \cdot g\) can be obtained from the graphs of \(f\) and \(g\) by multiplying corresponding \(y\)-coordinates.
• The graph of \(\dfrac{f}{g}\) can be obtained from the graphs of \(f\) and \(g\) by dividing the \(y\)-coordinates of \(f\) by the corresponding \(y\)-coordinates of \(g\) (as long as these \(y\)-values are non-zero).
• The behaviour of the graph of each (non-zero) function \(f\) and \(g\) will, in some way, influence the behaviour of the graphs of \(f \cdot g\) and \(\dfrac{f}{g}\).
• When both \(f\) and \(g\) are positive (above the \(x\)-axis), or both negative (below the \(x\)-axis), then \(f \cdot g\) is positive. When one of \(f\) or \(g\) is positive and the other is negative, then \(f \cdot g\) is negative. The zeros of \(f \cdot g\) occur when \(f(x) = 0\) or \(g(x) = 0\). The zeros of \(\dfrac{f}{g}\) occur when \(f(x) = 0\) and \(g(x) \neq 0\).
• For non-zero functions \(f\) and \(g\), if both \(f\) and \(g\) are even functions, or both \(f\) and \(g\) are odd functions, then \(f \cdot g\) and \(\dfrac{f}{g}\) are even functions. If one of \(f\) or \(g\) is an odd function and the other is an even function, then \(f \cdot g\) and \(\dfrac{f}{g}\) are odd functions.