Problem 81 - Precalculus Mathematics for Calculus

Question

Compositions Involving Trigonometric Functions This exercise explores the effect of the inner function \(g\) on a composite function \(y=f(g(x))\) (a) Graph the function \(y=\sin \sqrt{x}\) using the viewing rectangle \([0,400]\) by \([-1.5,1.5] .\) In what ways does this graph differ from the graph of the sine function? (b) Graph the function \(y=\sin \left(x^{2}\right)\) using the viewing rectangle \([-5,5]\) by \([-1.5,1.5] .\) In what ways does this graph differ from the graph of the sine function?

Step-by-Step Solution

Answer

(a) Periods stretch with increasing x; (b) Oscillations increase with |x|.

Step 1: Understanding the given function for part (a)

In part (a), we are analyzing the function \(y=\sin \sqrt{x}\). This function involves the sine function with an inner function \(g(x) = \sqrt{x}\), affecting how \(y\) behaves based on \(x\).

Step 2: Graphing the function for part (a)

Plot the graph of \(y=\sin \sqrt{x}\) in the range \(x \in [0,400]\) and \(y \in [-1.5, 1.5]\). Observe how the graph will start from an initial point at \(x=0\) and continues with increasing sine wave periods as \(x\) grows since \(\sqrt{x}\) increases at a decreasing rate.

Step 3: Identifying differences for part (a)

The graph of \(y=\sin \sqrt{x}\) starts at \(x=0\) rather than repeating every \(2\pi\). The periods of the sine function stretch as \(x\) increases since \(\sqrt{x}\) grows slower than \(x\).

Step 4: Understanding the given function for part (b)

In part (b), we analyze \(y=\sin(x^2)\). Here the inner function is \(g(x) = x^2\), affecting the output of the sine function significantly as \(x\) both increases and decreases from zero.

Step 5: Graphing the function for part (b)

Plot the graph of \(y=\sin(x^2)\) over the range \(x \in [-5,5]\) with \(y \in [-1.5, 1.5]\). Note how the periods of the sine function decrease; as \(x\) moves away from the origin, \(x^2\) grows quickly, leading to rapid oscillations.

Step 6: Identifying differences for part (b)

Unlike the standard sine function, \(y=\sin(x^2)\) has oscillations that increase in frequency as \(|x|\) becomes larger due to the rapid growth of \(x^2\). This creates denser wave patterns than the regular sine curve.

Key Concepts

Composite Functions

Composite functions involve the combination of two or more functions to form a new function. In this exercise, we deal specifically with trigonometric functions as composite functions, such as \(y = \sin \sqrt{x}\) and \(y = \sin(x^2)\). Here, the sine function is the outer function \(f\), and functions like \(\sqrt{x}\) or \(x^2\) are the inner functions \(g(x)\). These inner functions transform the input \(x\) before it is passed through the sine function.

When working with composite functions, it is essential to analyze how the inner function impacts the overall behavior of the composite function. For example, changing the inner function alters the frequency and period of the sine wave, resulting in unique graph characteristics. Understanding this relationship helps in graphing and analyzing how composite functions behave differently from standard trigonometric functions.

Graphing Techniques

Graphing trigonometric functions, especially composite ones, requires special techniques. When graphing \(y = \sin \sqrt{x}\), pay attention to how the graph starts at \(x=0\) rather than completing a full sine wave cycle. Initially, the wave lengthens as \(x\) increases, given that \(\sqrt{x}\) grows slower. This changes how many oscillations can be seen in a given range.

For \(y = \sin(x^2)\), note that as \(x\) moves away from zero, the value of \(x^2\) grows rapidly, causing the sine wave to oscillate more frequently. The graph will show these frequent oscillations in dense patterns. Graphing techniques involve choosing appropriate viewing windows and ranges that best display these variations, helping to capture distinct characteristics between the composite function and a standard sine curve.

Periodicity

Periodicity in trigonometric functions refers to how often the function repeats its values. For a standard sine function, the period is \(2\pi\), meaning it repeats every \(2\pi\) units.

In composite functions like \(y = \sin \sqrt{x}\), the period changes because \(\sqrt{x}\) grows more slowly than \(x\). This means that as \(x\) increases, the distance between the peaks of the waves increases, resulting in a stretched wave pattern.

Conversely, in \(y = \sin(x^2)\), the \(x^2\) component causes a rapid increase in function value, decreasing the period length. As a result, the waves become denser and occur more frequently. Understanding these variations in periodicity is pivotal for accurate graph interpretations and comparisons to the standard sine function.

Inner Functions

Inner functions in composite trigonometric functions significantly determine the output curve's nature. In our examples, \(\sqrt{x}\) and \(x^2\) serve as inner functions that modify the input before it's processed by the sine function.

• \(g(x) = \sqrt{x}\): This function increases at a decreasing rate, implying that as \(x\) becomes larger, it transforms more gradually. This affects the sine wave by lengthening the period as \(x\) becomes larger.
• \(g(x) = x^2\): On the other hand, this function grows rapidly especially as \(|x|\) increases, leading to more frequent oscillations in the sine wave. Hence, the graph displays rapid changes, diverging from the typical behavior of a simple sine function.

By comprehending the role of inner functions, students can predict and explain how transformations of the input translate into visible changes in the output graph.