Problem 79 - Algebra and Trigonometry

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) Frequency decreases with increasing x; (b) Frequency increases with deviation from x=0.

Step 1: Understanding the function y=sin(√x)

The function to graph is \( y = \sin(\sqrt{x}) \). For this function, the inner part \( g(x) = \sqrt{x} \) changes the input to \( \sin \), by taking the square root of \( x \). This affects the periodicity since \( g(x) \) spreads the inputs more widely before passing them to \( \sin \).

Step 2: Graphing y=sin(√x)

Using graphing tools or software, plot \( y = \sin(\sqrt{x}) \) over the domain \([0, 400]\) and range \([-1.5, 1.5]\). Observe that the graph starts at \( x=0 \), and as \( x \) increases, the frequency of oscillations decreases. Hence, this function has a decreasing frequency as \( x \) increases compared to regular \( \sin x \).

Step 3: Analyzing differences - y=sin(√x) vs y=sin(x)

In \( y = \sin(x) \), oscillations are uniform. In \( y = \sin(\sqrt{x}) \), the frequency decreases as \( x \) increases due to input adjustment by \( \sqrt{x} \). This means the waves stretch horizontally more in \( y = \sin(\sqrt{x}) \).

Step 4: Understanding the function y=sin(x²)

For \( y = \sin(x^2) \), the inner function \( g(x) = x^2 \) ramps up inputs faster as \( x \) moves away from zero. This affects the periodicity by increasing the frequency as inputs move further from zero.

Step 5: Graphing y=sin(x²)

Plot \( y = \sin(x^2) \) using the viewing window \([-5, 5]\) and range \([-1.5, 1.5]\). The graph will show a rising frequency of oscillations as \( x \) moves away from zero, with more pronounced peaks and valleys closer together.

Step 6: Analyzing differences - y=sin(x²) vs y=sin(x)

In \( y = \sin(x) \), oscillations are consistent. However, in \( y = \sin(x^2) \), frequency increases rapidly with \( |x| \) because the \( x^2 \) operation makes inputs grow faster, leading to rapid oscillations away from the origin.

Key Concepts

Composite Functions and Trigonometric Functions

In trigonometry, composite functions occur when a function is applied within another function. In our exercise, we are dealing with the composite functions like \( y = \sin(\sqrt{x}) \) and \( y = \sin(x^2) \). These are functions where a trigonometric function, \( \sin(x) \), is combined with another function, such as \( \sqrt{x} \) or \( x^2 \). This combination alters the behavior of the original trigonometric function.

• In \( y = \sin(\sqrt{x}) \), the inner function \( g(x) = \sqrt{x} \) modifies the input to the sine function by taking the square root. As a result, inputs are spaced more widely across the domain, causing the resulting graph to have a different horizontal stretching compared to \( \sin(x) \).


• In \( y = \sin(x^2) \), the inner function \( g(x) = x^2 \) speeds up the inputs, especially as \( x \) increases or decreases from zero. This causes the sine function to oscillate more frequently than in its standard form.

Certainly, understanding these inner functions, \( \sqrt{x} \) and \( x^2 \), is crucial as they control how the input values are transformed before reaching the trigonometric function. Composite functions offer a way to explore more complex behaviors and patterns in functions.

Graphing Techniques for Trigonometric Composites

To analyze functions like those in this exercise, efficient graphing techniques are vital. Graphing composite trigonometric functions allows us to visually understand how transformations affect the basic sine wave.

When graphing \( y = \sin(\sqrt{x}) \) over the domain of \([0, 400]\):

• The graph begins at the origin, \( x = 0 \), and stretches the oscillations as \( x \) increases. This shows a spreading effect where the frequency of oscillations reduces, presenting a different look than the standard sine curve.

Conversely, when graphing \( y = \sin(x^2) \) over the domain \([-5, 5]\):

• You observe that the sine waves become more concentrated and closer together. The frequency of oscillations increases as \( |x| \) moves away from zero, resulting in frequent wave peaks and troughs.

Mastering these graphing techniques is essential. They help to comprehend how the inner function’s transformation impacts the sine function's periodicity and frequency. With these graphs, we can immediately see the transformation effects and why they differ from the usual sine wave.

Understanding Periodicity in Composite Trigonometric Functions

Periodicity refers to a function's behavior repeating at regular intervals. The sine function, \( \sin(x) \), is known for its regular periodic nature with a period of \( 2\pi \). However, composite functions like \( y = \sin(\sqrt{x}) \) and \( y = \sin(x^2) \) demonstrate unique periodic behaviors due to their modified inputs.

In \( y = \sin(\sqrt{x}) \):

• The periodicity changes as the \( \sqrt{x} \) function alters the input spacing. The waves stretch out, meaning the period becomes less regular and decreases in frequency over increasing \( x \).

For \( y = \sin(x^2) \):

• The input grows quadratically, causing rapid oscillations as \( x \) deviates from zero, which increases the frequency of oscillations.

Understanding periodicity in these composite functions can be challenging but is critical for fully grasping their behavior. The periodic nature can be altered by transformations, giving each composite its own distinct pattern or rhythm, diverging from the classic sine curve. Recognizing these nuances aids in predicting and interpreting the function's long-term behavior.