Problem 78 - Precalculus: Functions and Graphs

Question

Use a graph to estimate the largest interval \([a, b],\) with \(a<0\) and \(b>0,\) on which \(f\) is one-to-one. $$f(x)=1.5 \cos \left(\frac{1}{2} x-0.3\right)+\sin (1.5 x+0.5)$$

Step-by-Step Solution

Answer

The largest interval is approximately ([ −0.5, 0.83]).

Step 1: Understand the Function

The given function is a combination of trigonometric functions, specifically the cosine and sine functions. These functions are periodic, meaning they repeat their values over regular intervals.

Step 2: Graph the Function

Using a graphing calculator or software, plot the function \(f(x) = 1.5 \cos \left(\frac{1}{2} x-0.3\right) + \sin (1.5 x+0.5)\). Look for the pattern and periodicity on the graph. Observe how the function behaves within different intervals.

Step 3: Identify Interval for One-to-One Function

A function is one-to-one if it passes the horizontal line test, which means any horizontal line intersects the graph at most once. Inspect the graph over different intervals, particularly focusing on ranges that span from \(a < 0\) to \(b > 0\), and identify the largest interval where this occurs.

Step 4: Estimate Interval [a, b]

Based on the graph, observe regions where the function rises or falls continuously without repeating any value. Typically, this interval could be one full period of the highest frequency component (here, determine by simplifying periodic components of sine and cosine). However, ensure that \(a < 0\) and \(b > 0\).

Step 5: Verify the Interval

Check your estimated interval by ensuring that it does not repeat on other cycles of the trig functions and that the values are unique in this range. Verify using the horizontal line test on the graph to confirm only one intersection per horizontal line.

Key Concepts

One-to-One Functions

A function is said to be one-to-one if it maps each value in its domain to a unique value in its codomain. This can be determined graphically by using the horizontal line test: if no horizontal line intersects the graph more than once, then the function is one-to-one.
For a trigonometric function, which is typically periodic, identifying intervals of one-to-oneness requires careful inspection.

• Trigonometric functions like sine and cosine inherently repeat their values due to their periodic nature.
• To identify an interval where the function is one-to-one, focus on a segment where the function either steadily increases or decreases.
• In periodic functions, this often means finding an interval that spans exactly one full cycle or period.
• It is crucial to ensure that the chosen interval satisfies the condition of the problem, such as having bounds with specific properties like \(a < 0\) and \(b > 0\).

Being able to visually assess these characteristics on a graph is an essential skill when determining one-to-oneness for complex functions.

Graphing Techniques

Graphing techniques serve as fundamental tools in understanding the behavior of mathematical functions, especially trigonometric ones. Proper graphing allows us to identify key features such as periodicity, amplitude, and phase shift.
When graphing a combination of trigonometric functions like \(f(x) = 1.5 \cos \left(\frac{1}{2} x-0.3\right) + \sin (1.5 x+0.5)\), it's important to consider the contribution of each component:

• Consider the amplitude, which determines the height of the graph from the midline (e.g., 1.5 for the cosine part).
• Examine the period of each function. For cosine, the period is \(\frac{2\pi}{\text{frequency}}\), here \(\text{frequency} = \frac{1}{2}\), giving a period of \(4\pi\). For sine, with frequency 1.5, the period is \(\frac{2\pi}{1.5}= \frac{4\pi}{3}\).
• Assess the phase shift, which indicates the horizontal displacement of the graph (e.g., -0.3 for the cosine and 0.5 for the sine).
• Use this information to plot several key points of the function, which can guide the overall shape and behavior of the graph.

Utilizing graphing software or a calculator can supplement manual efforts and offer a precise depiction of composite functions.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or cycles. Common examples include trigonometric functions like sine, cosine, and tangent.
Understanding the periodic nature of functions helps in determining properties like amplitude, frequency, and phase.

• Amplitude refers to the maximum height of the wave from its central axis; it determines how "tall" or "short" the wave appears.
• Frequency signifies how often the wave pattern repeats over a designated interval; higher frequency means more cycles in a given interval.
• Period is the length of one complete cycle of the function, usually calculated as \(\frac{2\pi}{\text{frequency}}\) for sine and cosine functions.
• Phase shift indicates a horizontal movement from the original position, showing how far the graph is shifted along the x-axis.

In the context of the given function, it is a blend of sine and cosine, making its behavior slightly more complex but still bound by these periodic principles. Therefore, identifying the period of each component and understanding how they combine is essential to analyze and interpret the overall function's behavior.