In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g.f(x) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)

Verified step by step guidance

1. Step 1: Identify the functions to be graphed. We have three functions: \( f(x) = \cos x \), \( g(x) = \sin 2x \), and \( h(x) = (f - g)(x) \).
2. Step 2: Graph \( f(x) = \cos x \) over the interval \( 0 \leq x \leq 2\pi \). This is a standard cosine wave starting at 1 when \( x = 0 \), reaching -1 at \( x = \pi \), and returning to 1 at \( x = 2\pi \).
3. Step 3: Graph \( g(x) = \sin 2x \) over the same interval. This function has a period of \( \pi \) (since the period of \( \sin kx \) is \( \frac{2\pi}{k} \)), so it completes two full cycles between \( 0 \) and \( 2\pi \).
4. Step 4: To find \( h(x) = (f - g)(x) = \cos x - \sin 2x \), subtract the y-coordinates of \( g(x) \) from \( f(x) \) at each point \( x \) in the interval \( 0 \leq x \leq 2\pi \).
5. Step 5: Plot the resulting points for \( h(x) \) on the same graph as \( f(x) \) and \( g(x) \). This will show how \( h(x) \) is derived from the combination of \( f(x) \) and \( g(x) \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental periodic functions that describe relationships between angles and sides in right triangles. The function f(x) = cos x represents the cosine function, which oscillates between -1 and 1, while g(x) = sin 2x is the sine function with a frequency that is double that of the standard sine function. Understanding these functions is crucial for graphing and analyzing their behavior over a specified interval.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the relationship between the input (x-values) and output (y-values). For the functions f(x) and g(x), their graphs will show how the cosine and sine functions behave over the interval from 0 to 2π. The ability to accurately graph these functions is essential for understanding how they interact when combined to form h(x).

Function Addition and Subtraction

Function addition and subtraction involve combining the outputs of two functions to create a new function. In this case, h(x) = (f - g)(x) means that for each x-value, we subtract the y-coordinate of g from the y-coordinate of f. This operation results in a new graph that reflects the differences between the two original functions, which is key to understanding the behavior of h(x) in relation to f(x) and g(x).