Problem 75 - Algebra and Trigonometry

Question

\(75-76\) 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) f is odd. (b) x-intercepts at x = 2kπ. (d) f(x)→0 as x→±∞. (e) Approaches 0 as x→0.

Step 1: Determine if the function is even or odd

A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). It is odd if it satisfies \( f(-x) = -f(x) \). If neither condition is true, the function is neither even nor odd.Calculate \( f(-x) = \frac{1- ext{cos}(-x)}{-x} = \frac{1- ext{cos}(x)}{-x} = -f(x) \).Thus, \( f(-x) = -f(x) \), indicating that the function \( f \) is odd.

Step 2: Find the x-intercepts

The \( x \)-intercepts occur where \( f(x) = 0 \). Set the function equal to zero: \[ \frac{1- ext{cos}x}{x} = 0 \].The numerator must be zero: \( 1 - \text{cos}x = 0 \) which simplifies to \( \text{cos}x = 1 \).Solving \( \text{cos}x = 1 \) gives \( x = 2k\pi \) where \( k \) is an integer. Therefore, the \( x \)-intercepts occur at \( x = 2k\pi \).

Step 3: Graph the function

Although the exact graph is not shown here, you would graph the function \( f(x) = \frac{1-\cos x}{x} \) using a graphing calculator or software. You would observe that there are vertical asymptotes at multiples of \( \pi \) due to the undefined points, and it approaches 0 as \( x \to \pm \infty \). Also, plot points near \( x = 0 \) for a better understanding of the behavior.

Step 4: Describe the function's behavior as \( x \rightarrow \pm\infty \)

As \( x \rightarrow \pm \infty \), the term \( \text{cos}x \) oscillates between -1 and 1. Thus, \( 1 - \text{cos}x \) oscillates between 0 and 2. Since \( x \) in the denominator increases indefinitely, \( \frac{1-\text{cos}x}{x} \) approaches zero. Therefore, the function approaches 0 as \( x \rightarrow \pm \infty \).

Step 5: Determine the behavior as \( x \rightarrow 0 \)

The function \( f(x) \) is not defined at \( x = 0 \) because it would involve division by zero. As \( x \to 0 \), \( 1 - \text{cos}x \approx \frac{x^2}{2} \) using the small angle approximation for cosine.Thus, \( f(x) \approx \frac{x^2/2}{x} = \frac{x}{2} \), which approaches 0 as \( x \to 0 \) from either side.

Key Concepts

x-intercepts

X-intercepts are the points where a function's graph crosses the x-axis, meaning these occur when the function equals zero. For the function \( f(x) = \frac{1-\cos x}{x} \) the x-intercepts can be found by setting the numerator equal to zero since the denominator cannot be zero (as dividing by zero is undefined). Therefore, we solve \( 1-\cos x = 0 \).

This equation simplifies to \( \cos x = 1 \), which holds true when \( x = 2k\pi \), where \( k \) is any integer. Essentially, this means the x-intercepts occur at multiples of \( 2\pi \). Understanding the points of x-intercepts helps in sketching the function's graph accurately as these are key details on the graph.

graphing functions

Graphing a function involves plotting its points to visualize its behavior. The function \( f(x) = \frac{1-\cos x}{x} \) can be challenging due to its characteristics, such as being undefined at \( x = 0 \) and having asymptotic behavior.

When graphing this function, use techniques like:

• Calculate and plot points around key points of interest, such as x-intercepts and where the function is undefined.
• Note the symmetry or pattern behavior: since \( f(x) \) is odd, the graph is symmetric about the origin.
• Identify trends: the function oscillates and approaches zero at infinity, but has vertical asymptotes where the denominator becomes zero.

Graphing software or calculators can aid in plotting these features accurately. The graph allows students to see where the function intercepts, asymptotes, and the overall behavior.

asymptotic behavior

Asymptotic behavior refers to how a function behaves as it approaches certain points, often towards infinity or where the function becomes undefined. For \( f(x) = \frac{1-\cos x}{x} \), it is defined everywhere except at x-values that make the denominator zero.

There are visible vertical asymptotes at multiples of \( \pi \), where \( x = k\pi \) (with non-zero k). At these points, the function becomes undefined, resulting in vertical lines on a graph that the function cannot cross.

Horizontal asymptotes are the lines that the graph approaches as \( x \rightarrow \pm \infty \). For \( f(x) \), the function approaches zero at infinity since the numerator oscillates between 0 and 2, while the denominator increases without bound. This behavior illustrates the end behavior of the function on a graph.

limits of functions

The limit of a function describes what the function's value approaches as the input nears a particular point. Calculating limits helps us understand the behavior of the function both at infinity and near undefined points.

For \( f(x) = \frac{1-\cos x}{x} \), the limit as \( x \rightarrow \infty \) is examined by noting that as \( x \) becomes very large, the function tends to zero.

When \( x \rightarrow 0 \), the function is undefined. However, using a small-angle approximation, \( 1-\cos x \approx \frac{x^2}{2} \), which allows for simplification: \( f(x) \approx \frac{x^2/2}{x} = \frac{x}{2} \). From this, we see it approaches zero both from the left and the right of zero. Understanding these limits clarifies how the function behaves in these critical regions and can explicate the graph's trend.