Problem 59
Question
\(59-60\) \(f, g,\) and \(f+g\) on a common screen to illustrategraphical addition. $$ f(x)=x, \quad g(x)=\sin x $$
Step-by-Step Solution
Verified Answer
Graphically add function values to create a combined plot.
1Step 1: Identify Functions
We have two functions given: 1. \( f(x) = x \) which is a straight line through the origin with slope 1.2. \( g(x) = \sin x \) which is a sinusoidal function, periodic with amplitude 1.
2Step 2: Sketch the Graph of \(f(x)\)
The graph of \( f(x) = x \) is a straight line passing through the origin (0,0) with a slope of 1. It increases linearly as \( x \) increases.
3Step 3: Sketch the Graph of \(g(x)\)
The graph of \( g(x) = \sin x \) is a wave that oscillates between 1 and -1 with a period of \( 2\pi \). It starts from (0,0), rises to (\( \frac{\pi}{2}, 1 \)), falls back to (\( \pi, 0 \)), dips to (\( \frac{3\pi}{2}, -1 \)), and returns to (\( 2\pi, 0 \)).
4Step 4: Determine \(f(x) + g(x)\)
The new function is \( f(x) + g(x) = x + \sin x \). To sketch this, add the corresponding y-values of \( f(x) \) and \( g(x) \) at each point \( x \).
5Step 5: Sketch the Graph of \(f(x) + g(x)\)
The graph of \( f(x) + g(x) \) will be a combination of the linear increase from \( f(x) \) and the oscillations from \( g(x) \). It will appear as a sinusoidal wave that shifts upwards as \( x \) increases linearly.
6Step 6: Interpret the Resulting Graph
By graphically adding \( f(x) \) and \( g(x) \), the resulting curve illustrates how a linear trend can combine with periodic oscillations, showing both linear growth and wave-like features in the sum.
Key Concepts
Linear FunctionsSinusoidal FunctionsFunction Addition
Linear Functions
Linear functions are the simplest type of functions, represented as a straight line on a graph. In mathematical terms, a linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept.
For the function \( f(x) = x \), the slope \( m \) is 1, which indicates that for every unit increase in \( x \), \( y \) also increases by one unit. There is no y-intercept (\( b = 0 \)), so the line passes directly through the origin (0,0).
Linear functions are important in mathematics because they provide a straightforward way to model relationships where change happens at a constant rate.
For the function \( f(x) = x \), the slope \( m \) is 1, which indicates that for every unit increase in \( x \), \( y \) also increases by one unit. There is no y-intercept (\( b = 0 \)), so the line passes directly through the origin (0,0).
Linear functions are important in mathematics because they provide a straightforward way to model relationships where change happens at a constant rate.
- They depict proportional relationships.
- Graphically represented as straight lines.
- Useful for understanding basic growth trends.
Sinusoidal Functions
Sinusoidal functions, like \( g(x) = \sin x \), are characterized by their wave-like patterns. These functions are defined by their amplitude, period, and phase shift. The amplitude is the peak value of the wave, which in \( g(x) = \sin x \) is 1, meaning the wave oscillates between 1 and -1.
The period, which is the distance required for the function to complete a full cycle, is \( 2\pi \) for the sine function. In simpler terms, a complete sine curve from start to finish, consisting of a rise and a fall, occurs between \( x = 0 \) and \( x = 2\pi \).
Sinusoidal functions are often used to model cyclic phenomena such as sound waves, light waves, and tides due to their repetitive nature.
The period, which is the distance required for the function to complete a full cycle, is \( 2\pi \) for the sine function. In simpler terms, a complete sine curve from start to finish, consisting of a rise and a fall, occurs between \( x = 0 \) and \( x = 2\pi \).
Sinusoidal functions are often used to model cyclic phenomena such as sound waves, light waves, and tides due to their repetitive nature.
- They have a regular, repeating pattern.
- Their graphs resemble smooth, continuous waves.
- Essential for modeling periodic behaviors.
Function Addition
Function addition involves summing two or more functions to produce a new function. In the context of graphical addition, we visually add the values of two functions at each point \( x \) to create a third function. In this exercise, we add \( f(x) = x \) and \( g(x) = \sin x \) to form \( f(x) + g(x) = x + \sin x \), combining the linear and sinusoidal functions into a single graph.
This combined graph reflects both the linear growth of the straight line and the oscillating nature of the sine wave. The resulting function showcases how a base level (linear trend) can be modulated by periodic fluctuations.
This combined graph reflects both the linear growth of the straight line and the oscillating nature of the sine wave. The resulting function showcases how a base level (linear trend) can be modulated by periodic fluctuations.
- Creates a new function through graphical addition.
- Combines characteristics of different function types.
- Helpful for understanding complex behavior arising from multiple influences.
Other exercises in this chapter
Problem 58
Length of a Shadow On a day when the sun passes directly overhead at noon, a six-foot-tall man casts a shadow of length $$ S(t)=6\left|\cot \frac{\pi}{12} t\rig
View solution Problem 59
Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sec t, \tan t ; \quad\) Quadrant II
View solution Problem 60
Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\csc t, \cot t ; \quad\) Quadrant III
View solution Problem 60
\(59-60\) \(f, g,\) and \(f+g\) on a common screen to illustrategraphical addition. $$ f(x)=\sin x, \quad g(x)=\sin 2 x $$
View solution