Suppose f(x)=cos x - sin x and g(x)=cos x + sin x. Explain why the graph of (fxg)(x) is equivalent to the graph of h(x)=cos x after it has been horizontally compressed by a factor of 1/2. Thanks so much...
To explain why the graph of (f(x) * g(x)) is equivalent to the graph of h(x) = cos(x) after it has been horizontally compressed by a factor of 1/2, we need to understand the concept of function composition and the properties of trigonometric functions.
Let's start by understanding the given functions f(x) = cos(x) - sin(x) and g(x) = cos(x) + sin(x). The function f(x) represents the difference between the cosine and sine functions, while g(x) represents their sum.
Now, let's consider the function (f * g)(x), which represents the product of f(x) and g(x). Mathematically, (f * g)(x) = f(x) * g(x) = (cos(x) - sin(x)) * (cos(x) + sin(x)).
To simplify this expression, we can use the formula (a - b)(a + b) = a^2 - b^2. Using this formula, we have:
(f * g)(x) = cos^2(x) - sin^2(x)
Now, let's recall the trigonometric identity cos^2(x) - sin^2(x) = cos(2x). So, we can rewrite (f * g)(x) as:
(f * g)(x) = cos(2x)
Here, we have obtained the function h(x) = cos(2x), which represents the graph of h(x) after a horizontal compression by a factor of 1/2.
By comparing h(x) = cos(2x) with the original function f(x) = cos(x) - sin(x), we can see that the graph of (f * g)(x) and h(x) are equivalent. The graph of (f * g)(x) is a horizontally compressed version of the graph of h(x) = cos(x) by a factor of 1/2.
So, the reason why the graph of (f * g)(x) is equivalent to the graph of h(x) = cos(x) after it has been horizontally compressed by a factor of 1/2 is due to the mathematical properties and identities of trigonometric functions.