Question

Graph the function using as many viewing rectangles as you need to depict the true nature of the function.

23. \(f\left( x \right) = \frac{{1 - \cos \left( {{x^4}} \right)}}{{{x^8}}}\)

Short answer

Graphs:

It can be observed that the graph is symmetric about \(y\)-axis. There is level off at about \(y = \frac{1}{2}\) for \(\left| x \right| = 0.7\). And there are smaller humps as \(\left| x \right|\) increases.

Step-by-step solution

Method to determine symmetry

Determine whether the function is symmetric or not by following any of the three steps:

1. The function is symmetric about \(y\)-axis if the function is even, and a function is even when \(f\left( { - x} \right) = f\left( x \right)\).
2. A function is symmetric about origin if it is an odd function, and a function is odd when \(f\left( { - x} \right) = - f\left( x \right)\).
3. The function is periodic when \(f\left( {x + p} \right) = f\left( x \right)\) for all \(x\) in domain, where \(p\) is a positive constant and it is known as the smallest period.

Determine symmetry

The given function is \(f\left( x \right) = \frac{{1 - \cos \left( {{x^4}} \right)}}{{{x^8}}}\).

For the given function \(f\left( x \right) = \frac{{1 - \cos \left( {{x^4}} \right)}}{{{x^8}}}\), \(f\left( { - x} \right) = x\), so the function is symmetric about \(y\)-axis.

Draw the graph of the given function

To check the answer, visually draw the graph of the function \(f\left( x \right) = \frac{{1 - \cos \left( {{x^4}} \right)}}{{{x^8}}}\) by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\left( {1 - \cos \left( {{X^4}} \right)} \right)/{X^8}\)in\({Y_1}\)the tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of \(f\left( x \right)\) is shown below:

This graph shows that there is a level off at about\(y = \frac{1}{2}\)for \(\left| x \right| = 0.7\).Then the graph drops down to the \(x\)-axis.

Use the “ZOOM” option in the graphing calculator for the interval \(2 < x < 3\).

This graph shows that, \(\left| x \right|\) increases, where the humps of the function become smaller as \(\left| x \right|\) increases.