Problem 57 - Algebra and Trigonometry

Question

Graph \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=x, \quad g(x)=\sin x $$

Step-by-Step Solution

Answer

Graph \(f(x) = x\), \(g(x) = \sin x\), and their sum \(f(x) + g(x) = x + \sin x\) on the same plane.

Step 1: Understand the Functions

We are given the functions \(f(x) = x\) and \(g(x) = \sin x\). The function \(f(x) = x\) represents a straight line with slope 1 and passes through the origin. The function \(g(x) = \sin x\) is a periodic wave that oscillates between -1 and 1.

Step 2: Graph the Function f(x)

Plot the function \(f(x) = x\) on a coordinate plane. This will be a straight line starting from the origin and extending diagonally in both the positive and negative quadrants at a 45-degree angle.

Step 3: Graph the Function g(x)

Plot the function \(g(x) = \sin x\) on the same coordinate plane. This is a sine wave that starts at the origin (0,0), reaches a maximum at \(\frac{\pi}{2}\), returns to zero at \(\pi\), reaches a minimum at \(\frac{3\pi}{2}\), and returns to zero at \(2\pi\), continuing this pattern periodically.

Step 4: Graph the Function f(x) + g(x)

To graph \(f(x) + g(x)\), compute the sum at various values of \(x\). For instance, at \(x = 0\), \(f(0) + g(0) = 0 + 0 = 0\). At \(x = \frac{\pi}{2}\), \(f\left(\frac{\pi}{2}\right) + g\left(\frac{\pi}{2}\right) = \frac{\pi}{2} + 1\). Continue this for enough points to capture the behavior of the sum function, then plot these points to visualize the new waveform.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions you can come across in mathematics. They are in the form of \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the specific case, the function \(f(x) = x\) is a linear function with a slope of 1 and a y-intercept of 0. This means it passes through the origin (0,0), making it a special type of line known as a proportional relationship.
Linear functions are always straight lines. The steepness of the line is determined by the slope \(m\). In this exercise, the slope is 1, meaning that for every increase of 1 in \(x\), \(f(x)\) increases by 1 as well.
Key characteristics of linear functions include:

• Constant rate of change.
• Symmetrical about the origin if the form is \(f(x) = mx\).
• Graphing results in a straight line.

Understanding these properties helps when graphing and adds clarity when they are combined with other types of functions, like the sine wave in this exercise.

Sine Wave

The sine wave is a key player in trigonometry and often appears in physics, engineering, and signal processing. It is defined by the function \(g(x) = \sin x\), which describes a repetitive oscillating pattern.
The basic sine wave starts at the origin (0,0), reaches its first maximum point at \(\frac{\pi}{2}\), where the value is 1, descends back to 0 at \(\pi\), hits a minimum at \(\frac{3\pi}{2}\), where the value is -1, and returns to 0 again at \(2\pi\). This cycle repeats indefinitely beyond the visible graph.
Sine waves have distinct properties:

• Amplitude: The peak height from the center line (here it is 1, indicating how far g(x) can reach above and below the x-axis).
• Period: The length it takes to complete one full cycle (in this case, \(2\pi\)).
• Frequency: The number of cycles that occur in a unit interval (often related to 1 over the period).

When you graph a sine wave, you notice its smooth, wave-like structure which makes it unique and useful for modeling oscillations in real-world scenarios.

Function Plotting

Function plotting involves visually representing mathematical functions on a coordinate plane. This helps in easily understanding the behavior and characteristics of functions. Plotting functions like \(f(x) = x\) and \(g(x) = \sin x\) on the same graph reveals how different functions can interact with each other.
The function \(f(x) + g(x)\) represents the graphical addition of these two functions. This process involves plotting the resulting function that comes from adding their y-values at corresponding x-values. For example, at \(x = 0\), both functions equal 0, so \(f(0) + g(0) = 0\), and at \(x = \frac{\pi}{2}\), \(f(x)\) equals \(\frac{\pi}{2}\) and \(g(x)\) equals 1, resulting in \(\frac{\pi}{2} + 1\).
Important steps in function plotting include:

• Choose a range of x-values that capture essential behavior of functions.
• Calculate corresponding y-values.
• Mark these points on the coordinate plane.
• Connect the points, observing patterns or curves exhibited by the plotted lines.

In the case of \(f(x) + g(x)\), the plotted result will visually display a balance between the linear growth and the oscillating nature of the sine wave. This helps in understanding how the combined function behaves.