Determination of Domain and Range for the Function y cos^{-1}(2x/1x)

Understanding the domain and range of a function is crucial for its proper evaluation and application. This article will explore the domain and range of the function y cos^{-1}left(frac{2x}{1x}right). We will delve into the properties of the inverse cosine function and the constraints that frac{2x}{1x} must satisfy.

Introduction to the Function

The function cos^{-1}(y) is defined for y in the interval [-1, 1]. This property is fundamental to our analysis of the given function, y cos^{-1}left(frac{2x}{1x}right).

Determining the Domain

To find the domain of the function, we need to determine the interval for x such that the expression frac{2x}{1x} lies within the domain of the inverse cosine function, i.e., [-1, 1].

Step-by-Step Analysis

Let's set up the inequality:

-1 leq frac{2x}{1x} leq 1

1. Left Inequality:

-1 leq frac{2x}{1x}

Rearranging, we get:

-11x leq 2x

-1 - x leq 2x

-1 leq 3x

x geq -frac{1}{3}

2. Right Inequality:

frac{2x}{1x} leq 1

Rearranging, we get:

2x leq 1 x

x leq 1

Combining the Inequalities:

The domain of x is the interval where both inequalities hold true:

-frac{1}{3} leq x leq 1

Determining the Range

Once we have the domain, we can determine the range of the function y cos^{-1}left(frac{2x}{1x}right). We will evaluate the function at the endpoints of the domain to determine the range.

Endpoint Analysis

For x -frac{1}{3}:

frac{2(-frac{1}{3})}{1 - (-frac{1}{3})} frac{-frac{2}{3}}{frac{2}{3}} -1

y cos^{-1}(-1) pi

For x 1:

frac{2(1)}{1 1} frac{2}{2} 1

y cos^{-1}(1) 0

Determining the Range

Since the cosine function is decreasing in the interval [0, pi], as x goes from -frac{1}{3} to 1, y decreases from pi to 0.

Therefore, the range of the function is: [0, pi]

Conclusion

In summary, the domain of the function y cos^{-1}left(frac{2x}{1x}right) is -frac{1}{3} leq x leq 1 and the range is [0, pi].