Problem 55 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$f(x)=2 \cos 3 x$$.

Step-by-Step Solution

Answer

The domain of the function \(f(x) = 2\cos 3x\) is all real numbers (-∞, ∞) and the range is [-2, 2].

Step 1: Determine the Domain

The x-values (or inputs) for a cosine function can be any real number. Therefore, the domain is all real numbers, represented as (-∞, ∞).

Step 2: Determine the Range

The range of the function is the set of possible output values. A standard cosine function oscillates between -1 and 1. Since the function here is \(2\cos 3x\), the range is magnified by a factor of 2 which impacts the maximum and minimum values. This means that instead of ranging from -1 to 1, the function will range from -2 to 2.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and ratios of right triangle sides. They are incredibly useful in various fields such as physics, engineering, and computer graphics.
One of the most common trigonometric functions is the cosine function, which we'll explore further later. Trigonometric functions have unique properties:

• They are periodic, meaning they repeat their values in regular intervals.
• Their graphs form wave-like patterns, known as sine and cosine waves for sine and cosine functions respectively.
• These functions are essential in modeling cyclic phenomena such as sound waves, light, and tides.

Understanding trigonometric functions allows us to analyze and predict patterns and behaviors in natural and engineered systems.

Cosine Function

The cosine function is a key trigonometric function that provides the horizontal coordinate of a point on the unit circle as the angle increases from the positive x-axis.
It is represented by the equation:\[ ext{cos}(x) \]
where 'x' is the angle measured in radians.

• The graph of the basic cosine function, \( \cos(x) \), oscillates between -1 and 1.
• It completes one full cycle every 2π radians, which is its period.
• This cyclical nature makes it ideal for modeling phenomena that repeat over time.

The cosine function can also be transformed through scaling, shifting, and stretching, which alters its amplitude and frequency.

Real Numbers

Real numbers encompass all the numbers we use in daily life, including rational and irrational numbers. They form the set of numbers that cover the entire number line, including integers, fractions, and decimals.
In the context of trigonometric functions such as the cosine function, real numbers are crucial:

• The domain of standard trigonometric functions, such as \( \cos(x) \), includes all real numbers, represented by the interval \( (-\infty, \infty) \).
• This means you can input any real number into a cosine function and get a real number as output.

Understanding real numbers helps grasp the extensive applicability of functions like cosine, which are not limited to discrete or whole number inputs.

Function Transformation

Function transformation involves adjusting the basic shape and position of a function's graph. These transformations include scaling, shifting, or flipping the graph to fit certain specifications or model real-world situations better.
With our specific example of the function \( 2 \cos 3x \), several transformations have occurred:

• The coefficient 2 stretches the graph vertically, changing its amplitude from 1 to 2, so it ranges from -2 to 2.
• The coefficient 3 inside the cosine argument reduces the period of the function from \(2\pi\) to \(\frac{2\pi}{3}\), making the graph complete its cycles more frequently.

These transformations alter how we perceive and calculate the range and behavior of the function, making it essential to understand how each parameter affects the original function.