Problem 58 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$h(x)=\sqrt{\cos ^{2} x}$$.

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\sqrt{\cos ^{2} x}\) is all real numbers, and the range is \([0, 1]\).

Step 1: Analyze the Function

The function given is \(h(x) = \sqrt{\cos^2(x)}\). The function is made up of the square root of squared cosine function. The cosine function, \(\cos{x}\), is defined for all real numbers, hence the square of cosine function, \(\cos^2{x}\), is also defined for all real numbers. Considering that square of any real number is always nonnegative, we know the square root function will be defined for the resultant values of \(\cos^2{x}\). This confirms that the domain of \(h(x)\) is all real numbers.

Step 2: Determine the Range

To find the range, we need to analyze the values \(\cos^2{x}\) can take. Cosine of any real number is always in the interval \([-1, 1]\). However, when squared, cos function produces values in \([0, 1]\) interval. As square root function is applied to these values, the range of the function also remains in the \([0, 1]\) interval.

Key Concepts

Domain of a Function

The domain of a function refers to the complete set of values for which the function is defined. In simpler terms, it is the collection of "input" values that you can put into the function without encountering any undefined mathematical situations. For the given function, \( h(x) = \sqrt{\cos^2(x)} \), understanding the nature of its components is crucial.

• The cosine function, \( \cos(x) \), itself is defined for all real numbers.
• Squaring \( \cos(x) \) results in \( \cos^2(x) \), which remains defined for all real numbers because it transforms any real number \( x \) into a nonnegative value.
• The square root operation on \( \cos^2(x) \) will also be defined for these nonnegative values.

Thus, the domain of \( h(x) \) is all real numbers, or in interval notation: \(( -\infty, \infty )\). This means that you can substitute any real number for \( x \) in the function and have it yield a real value.

Range of a Function

The range of a function is the set of possible "output" or "function values" that the function can produce. To find the range of \( h(x) = \sqrt{\cos^2(x)} \), let’s break down the values \( \cos^2(x) \) can reach and the effects of the square root.

• First, consider \( \cos(x) \): It always lies within the interval \( [-1, 1] \) for any real number \( x \).
• Squaring this results in \( \cos^2(x) \), which therefore ranges within \([0, 1]\), as any negative becomes positive on squaring.
• Finally, taking the square root of \( \cos^2(x) \) doesn’t change this range but reaffirms it to be within \([0, 1]\).

Thus, the range of \( h(x) \) is \([0, 1]\). Whether \( x \) varies continuously over its domain, the function output persists within these bounds, never producing a negative number or one exceeding unity.

Cosine Function

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function which plays a vital role in various applications, from modeling waves to solving triangles. Let's understand the key aspects that are relevant here.

• Periodicity: \( \cos(x) \) is periodic with a period of \( 2\pi \), meaning its pattern of values repeats every \( 2\pi \, \text{radians} \).
• Range of Values: The function value lies between \([-1, 1]\), meaning any input can only produce outputs within this band.
• Even Function: It is symmetrical about the y-axis, or mathematically, \( \cos(-x) = \cos(x) \).

These properties make \( \cos(x) \) particularly useful in physics and engineering fields where natural periodic phenomena occur. The ability of \( \cos(x) \) to generate predictable outputs enables accurate modeling and analysis.

Square Root Function

The square root function is one of the basic operations in mathematics, mostly denoted by \( \sqrt{x} \). It finds extensive use due to its intuitive nature of finding a number which, when multiplied by itself, results in \( x \). Let's delve into aspects relevant to this function.

• Definition Domain: The square root function is defined only for nonnegative numbers. Hence, only inputs \( x \geq 0 \) produce real number outputs.
• Output Behavior: The values are non-negative as well, and the function increases monotonically—meaning, greater inputs yield higher outputs.
• Not Entirely Reversible: While squaring can be reversed by taking the square root, negative roots don't have real square roots in this basic context.

Thus, in the given function \( h(x) = \sqrt{\cos^2(x)} \), it helps ensure that every transformation yields a feasible real number, due to \( \cos^2(x) \) being always nonnegative.