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. Please help, thank you!
(fxg)(x) = (cosx - sinx)(cosx + sinx)
= cos^2 x - sin^2 x
= cos 2x , by definition.
notice the period of the new one is 2π/2 = π
while the period of the first one was 2π
notice the compression by a factor of 1/2 ?
Oh your right! Now I see that, thanks so much!
To understand why the graph of (f(x) * g(x)) is equivalent to the graph of h(x) after being horizontally compressed by a factor of 1/2, we need to analyze the properties and behavior of the functions involved.
1. Start by expanding (f(x) * g(x)):
(f(x) * g(x)) = (cos x - sin x) * (cos x + sin x)
= cos²x - sin²x
2. Recall the trigonometric identity cos²x - sin²x = cos(2x):
(f(x) * g(x)) = cos(2x)
3. Now, consider the function h(x) = cos x.
If we compare h(x) and cos(2x), we can notice that the argument of the cosine function is doubled in cos(2x) compared to h(x).
4. The horizontal compression by a factor of 1/2 means that the argument of the cosine function is multiplied by 1/2, in other words, x is replaced by (1/2)x.
Therefore, h((1/2)x) = cos((1/2)x).
5. Substituting (1/2)x in the expression for cos(2x), we get:
cos((1/2)x * 2) = cos(x)
6. Simplifying further, we have:
cos(x) = cos(x)
7. Thus, (f(x) * g(x)) and h(x) = cos x are equivalent when the former is horizontally compressed by a factor of 1/2.
In summary, the graph of (f(x) * g(x)) = cos(2x) is equivalent to the graph of h(x) = cos x after being horizontally compressed by a factor of 1/2.