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Chapter 1: Problem 60

\(g(x)=|x| \cos x\)

Short Answer

Expert verified
The function \(g(x) = |x| \cos x\) represents the graph of \(x \cos x\) for \(x \geq 0\) and its reflection about the y-axis for \(x < 0\). The reflection is due to the function becoming \( -x \cos x \) for \(x < 0\).

Step by step solution

01

Analyze the absolute value and cosine function

Understand that the cosine function oscillates between -1 and 1 and its graph is symmetrical around the y-axis. The absolute function, on the other hand, takes the absolute value of any number, effectively making any negative number into a positive number. Since \(|x|\) is multiplied by \(\cos(x)\), the sign flipping properties of \(\cos(x)\) for negative x-values will not be observed in the overall function.
02

Split the function into two intervals

Due to the absolute value, split the function into two intervals: when \(x < 0\) and when \(x \geq 0\). For \(x<0\), the function becomes \(g(x) = -x \cos(x)\) because |x| is -x for this interval. For \(x \geq 0\), the function remains the same, \(g(x) = x \cos(x)\), because |x| is x for this interval.
03

Reflect the graph of \( \cos(x) \) on y-axis

Finally, the function \(g(x)\) can be visualized as the graph of \(x\cos(x)\), reflected on the y-axis for the interval \(x<0\) because of the negative sign now included in the function for this interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is a fundamental trigonometric function that appears frequently in mathematics, particularly in the study of waves, oscillations, and circles. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function, denoted as \(\cos(x)\), exhibits a wave-like pattern as it oscillates between -1 and 1 on a graph.

When graphing the cosine function, it starts at its maximum value of 1 when \(x=0\) and proceeds to decrease until it reaches -1 at \(\pi\) radians (or 180 degrees), showing that it is an even function, which means it's symmetric about the y-axis. Understanding the behavior of the cosine function is crucial before applying transformations, such as the absolute value.
Function Transformation
Function transformation involves modifying the original function in various ways 鈥� like shifting, stretching, compressing, or reflecting it 鈥� to create a new function. In our exercise, the cosine function undergoes a transformation by being multiplied by the absolute value of \(x\). This operation affects the graph of the cosine function significantly.

When the absolute value function interacts with the cosine function, \(g(x)=|x| \cos x\), the original symmetry of \(\cos(x)\) is altered. For \(x\) values greater than or equal to zero, the function remains \(x\cos(x)\), showing no change. However, for negative \(x\) values, the effect of \(|x|\) is to reflect the cosine function about the y-axis, thereby eliminating negative \(x\) values and flipping the waves of the cosine function in the negative x-region upward.
Graphing Absolute Value Functions
Graphing absolute value functions introduces a unique visual feature: the 'V' shape, which is a result of converting all negative input values to their positive counterparts. The graph of \(g(x)=|x|\cos(x)\) contains elements of both the V-shape and the oscillating waves of the cosine function.

To accurately graph this function, one must consider the separate intervals for negative and positive values of \(x\). For positive \(x\), the graph looks like waves that increase in amplitude as \(x\) increases. Conversely, for negative \(x\), the graph is the mirror image of the positive side, with waves reflected over the y-axis due to the absolute value. This results in an oscillating pattern that grows outward from the y-axis as \(x\) becomes more negative, embodying both the properties of absolute value functions and the cosine function.

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Most popular questions from this chapter

From city \(A\) to city \(B\), a plane flies 650 miles at a bearing of \(48^{\circ}\). From city \(B\) to city \(C\), the plane flies 810 miles at a bearing of \(115^{\circ}\). Find the distance from city \(A\) to city \(C\) and the bearing from city \(A\) to city \(C\).

In Exercises 89 and 90, write the function in terms of the sine function by using the identity $$ A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right) $$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$ f(t)=3 \cos 2 t+3 \sin 2 t $$

Describe the behavior of \(f(\theta)=\sec \theta\) at the zeros of \(g(\theta)=\cos \theta\). Explain your reasoning.

The formulas for the area of a circular sector and arc length are \(A=\frac{1}{2} r^{2} \theta\) and \(s=r \theta\), respectively. ( \(r\) is the radius and \(\theta\) is the angle measured in radians.) (a) For \(\theta=0.8\), write the area and arc length as functions of \(r\). What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more rapidly as \(r\) increases. Explain. (b) For \(r=10\) centimeters, write the area and arc length as functions of \(\theta\). What is the domain of each function? Use a graphing utility to graph and identify the functions.

Use a graphing utility to graph the function. $$ f(x)=-\arcsin 2 x $$
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