Problem 73 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Is \(f\) even, odd, or neither? (b) Find the \(x\) -intercepts of the graph of \(f\) (c) Graph \(f\) in an appropriate viewing rectangle. (d) Describe the behavior of the function as \(x \rightarrow \pm \infty\) (e) Notice that \(f(x)\) is not defined when \(x=0 .\) What happens as \(x\) approaches 0\(?\) $$ f(x)=\frac{1-\cos x}{x} $$

Step-by-Step Solution

Answer

(a) Odd; (b) \(x = 2n\pi\) for \(n \neq 0\); (d) \(f(x) \to 0\) as \(x \to \pm\infty\); (e) As \(x \to 0\), \(f(x) \to 0\).

Step 1: Determine if the Function is Even, Odd, or Neither

To determine if the function \( f(x) \) is even, odd, or neither, recall the definitions: A function is even if \( f(-x) = f(x) \) for all \( x \). A function is odd if \( f(-x) = -f(x) \) for all \( x \). Substitute \( -x \) into \( f(x) \):\[ f(-x) = \frac{1 - \cos(-x)}{-x} = \frac{1 - \cos x}{-x} = -\frac{1 - \cos x}{x} = -f(x) \]Since \( f(-x) = -f(x) \), the function \( f(x) \) is odd.

Step 2: Find the x-Intercepts

The \( x \)-intercepts of the function are the values of \( x \) for which \( f(x) = 0 \). Set \( f(x) \) to zero:\[\frac{1 - \cos x}{x} = 0 \]For the fraction to be zero, the numerator must be zero without causing the denominator to also be zero. Hence,\[ 1 - \cos x = 0 \rightarrow \cos x = 1 \]The solutions to \( \cos x = 1 \) are \( x = 2n\pi \) where \( n \) is an integer. Therefore, the potential \( x \)-intercepts are at \( x = 2n\pi \), but we exclude \( x = 0 \) since \( f(x) \) is not defined there.

Step 3: Graph the Function

When graphing \( f(x) = \frac{1 - \cos x}{x} \), consider that the function is not defined at \( x=0 \). The graph will exhibit gaps or asymptotic behavior at these points. Use software or graphing calculator for accuracy, ensuring the x-axis includes multiples of \( 2\pi \), excluding zero. The function oscillates around zero with symmetrical dips due to its odd nature.

Step 4: Describe the End Behavior of the Function

As \( x \to \pm\infty \), the cosine function oscillates between -1 and 1. Therefore, the behavior of\[f(x) = \frac{1 - \cos x}{x}\]tends towards zero because the numerator fluctuates between 0 and 2 while the denominator grows large in magnitude, leading to increasingly smaller fractions.

Step 5: Analyze the Behavior as x Approaches Zero

As \( x \rightarrow 0 \), \( f(x) \) approaches an indeterminate form \( \frac{0}{0} \). Applying L'Hôpital's Rule yields a more straightforward expression to analyze:\[\lim_{x\to0} \frac{1 - \cos x}{x} = \lim_{x\to0} \frac{\sin x}{1} = 0\]Thus, \( f(x) \) approaches zero as \( x \) approaches zero, even though \( f(x) \) is not defined exactly at \( x = 0 \).

Key Concepts

X-Intercepts

X-intercepts are the points where the graph of a function crosses the x-axis. When a function equals zero at a particular value of x, that's an x-intercept. For our function, we set \[ f(x) = \frac{1 - \cos x}{x} = 0 \]and solve for x.

The numerator dictates the x-intercepts, leading us to find when \[ 1 - \cos x = 0 \]Thus, \[ \cos x = 1 \]
The solutions are at each \( x = 2n\pi \)(where \( n \) is an integer) because these are the angles where the cosine of x equals one. Remember, the function is not defined at \( x = 0 \), so we don't consider that as an intercept.

These x-intercepts occur at evenly spaced intervals on the graph, illustrating the repetitive nature of trigonometric functions. Understanding x-intercepts is crucial when analyzing the roots of functions and predicting their behavior.

Function Graphing

Graphing a function helps us visualize its behavior. For the function \[ f(x) = \frac{1 - \cos x}{x} \], we need to consider a few key aspects to draw it accurately. Firstly, notice that the function is not defined at \( x = 0 \).

To construct the graph:

• Plot x-intercepts at \( x = 2n\pi \), excluding \( x = 0 \).
• Recognize that the function is odd (\( f(-x) = -f(x) \)). This symmetry means the graph has rotational symmetry around the origin.
• Use a graphing calculator for precision, especially for oscillations.

The graph exhibits wave-like fluctuations due to the periodic nature of cosine. Each wave symmetric about the origin adds up to the complexity while tracing this function. Function graphing lets us visually interpret how functions behave with respect to inputs (x-values).

End Behavior

End behavior helps us understand how functions behave as inputs grow significantly large or small. For our function, \[ f(x) = \frac{1 - \cos x}{x} \], we observe what happens as x approaches infinity.

As \( x \rightarrow \pm\infty \), the denominator becomes very large, while the numerator remains bounded between 0 and 2 (due to cosine varying only between -1 and 1). This results in:

• The fraction getting smaller and smaller.
• Thus, \( f(x) \rightarrow 0 \) as \( x \rightarrow \pm\infty \).

This shows the function touches the x-axis asymptotically, describing its ultimate fate through visual and computational means. Understanding end behavior equips us with insights about the function beyond its typical domain.

L'Hôpital's Rule

L'Hôpital's Rule is a handy calculus tool for evaluating limits that initially result in indeterminate forms, like \( \frac{0}{0} \). For our function, \( f(x) = \frac{1 - \cos x}{x} \), as \( x \rightarrow 0 \), the function is not defined, but we can explore its limit.

With the indeterminate form \( \frac{0}{0} \), apply L'Hôpital's Rule by differentiating the numerator and denominator:\[ \lim_{x \to 0} \frac{1 - \cos x}{x} = \lim_{x \to 0} \frac{\sin x}{1} \]We simplify further to see:

Since \( \sin x \rightarrow 0 \) when \( x \rightarrow 0 \), the limit evaluates to 0.
Therefore, \( f(x) \rightarrow 0 \) as \( x \rightarrow 0 \) even though the function is undefined at this exact point. Understanding L'Hôpital's Rule is critical for navigating functions that involve limits of quotients. It provides a way to process and comprehend otherwise complex behavior at critical points.