Problem 54 - Calculus: One and Several Variables

Question

Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=\sqrt{1-x^{2}}, \quad g(x)=\sin 2 x$$

Step-by-Step Solution

Answer

The composition \(f(g(x)) = \sqrt{1-(\sin 2x)^{2}}\) has domain \(\(-\infty < x < \infty\) \), and the composition \(g(f(x)) = \sin 2\sqrt{1 - x^{2}}\) has domain \(\(-1 \leq x \leq 1\).

Step 1: Compute the Composition \(f \circ g\)

The composition \(f \circ g\) means we are plugging \(g(x)\) into \(f(x)\). So we get \(f(g(x))=f(\sin 2x)=\sqrt{1-(\sin 2x)^{2}}\). Note that because the sine function only gives values between -1 and 1, the entire domain of x is valid for this composition.

Step 2: Compute the Composition \(g \circ f\)

Similarly, to compute the composition \(g \circ f\), we plug \(f(x)\) into \(g(x)\). We get \(g(f(x))=g(\sqrt{1 - x^{2}})=\sin 2\sqrt{1 - x^{2}}\). Because the outputs of the square root function are only positive numbers or zero, there is no problem with the domain when plugging \(f(x)\) into \(g(x)\), therefore the domain matches that of \(f(x)\), which are numbers between -1 and 1 inclusive.

Step 3: Summary of the Domains

The domain of \(f(g(x))\) is \(\(-\infty < x < \infty\) \), all real numbers, because there are no restrictions on the input of the sine function, and the resulting output still falls within the input requirements of the square root function. The domain of \(g(f(x))\) is \(\(-1 \leq x \leq 1\) \), since when forming this composition, the domain restrictions of the square root function are preserved.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when dealing with function compositions. A function's domain is the set of all possible input values (typically represented by \(x\)) for which the function is defined. When composing functions, like with \(f \circ g\), the domain must satisfy the restrictions of both functions involved.
For example, in the original problem:

• \(f(x) = \sqrt{1-x^2}\): This function is only defined if \(1-x^2 \geq 0\). Solving this inequality, we get the domain of \(f\) as \(-1 \leq x \leq 1\).
• \(g(x) = \sin 2x\): The sine function has a domain of all real numbers, \((-\infty, \infty)\), allowing any real \(x\) as input.

When determining the domain for \(f \circ g\), we check that \(g(x)\) yields values suitable for \(f\), resulting in the overall domain of \(f(g(x))\) being all real numbers.
On the other hand, for \(g \circ f\), since \(f(x)\) imposes \(-1 \leq x \leq 1\), this defines the domain for \(g(f(x))\). Thus, understanding and combining domains ensures valid compositions in mathematics.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are frequently encountered in calculus and function composition. They have periodic properties and are defined for all real numbers, which is essential for calculating compositions.
Let's explore the relevant properties of the sine function:

• Range: The sine function produces values between \(-1\) and \(1\) inclusive. In our context, \(g(x) = \sin 2x\) multiplies the input by 2, but it keeps the range within these boundaries.
• Periodicity: Sine functions repeat every \(2\pi\) radians. In \(g(x)\), the period is halved to \(\pi\) due to the factor of 2.
• Smooth and Continuous: Sine functions are smooth and continuous, making them easy to use in composition, as these properties guarantee a straightforward transition between function domains when forming compositions like \(f \circ g\) and \(g \circ f\).

As trigonometric functions, their behavior directly influences the overall domain and output of compositions, as we observed with \(g(x)\) fitting smoothly into \(f(x)\). Understanding these properties helps predict and manipulate function outputs effectively.

Square Root Function

The square root function is another cornerstone in function composition, transforming input values under specific conditions.
Key features to consider when dealing with a square root function such as \(f(x) = \sqrt{1 - x^2}\):

• Domain: The expression \(1-x^2\) must be non-negative, confining \(x\) to the interval \([-1, 1]\). This smooth restriction ensures the resulting values are suitable for further operations, such as composition with other functions.
• Range: The output of the square root function is always non-negative, ranging from 0 to 1 when considering \(f(x)\). This simplifies understanding of output when forming compositions like \(g(f(x))\).
• Continuity: Square root functions are continuous within their domains, ensuring no discontinuities or gaps exist, which might disrupt calculations or graphical representations.

When combined in compositions, the square root function's inherent properties, such as non-negativity and domain restrictions, play a pivotal role in determining the nature and behavior of the composite function. Appreciating these attributes is essential for successful evaluations and graphing.