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 understand why the graph of (f ⨉ g)(x) is equivalent to the graph of h(x)=cos x after being horizontally compressed by a factor of 1/2, let's break it down step by step.
1. Begin with the functions f(x)=cos x - sin x and g(x)=cos x + sin x.
2. The expression (f ⨉ g)(x) denotes the product of the two functions: f(x) multiplied by g(x). In other words, (f ⨉ g)(x) = f(x) ⨉ g(x).
3. Expanding the expression, we have (f ⨉ g)(x) = (cos x - sin x) ⨉ (cos x + sin x).
4. Applying the distributive property, we can multiply the terms within the parentheses: (f ⨉ g)(x) = (cos x ⨉ cos x) + (cos x ⨉ sin x) + (-sin x ⨉ cos x) + (-sin x ⨉ sin x).
5. Simplifying the expression further, we obtain: (f ⨉ g)(x) = cos²x + cos x ⨉ sin x - sin x ⨉ cos x - sin²x.
6. Notice that the term cos x ⨉ sin x and -sin x ⨉ cos x are equal but have opposite signs. Therefore, their sum is zero. Simplifying, we have: (f ⨉ g)(x) = cos²x - sin²x.
7. Recall the trigonometric identity cos²x - sin²x = cos(2x).
8. So, we conclude that (f ⨉ g)(x) = cos(2x).
9. Now, let's consider the function h(x) = cos x.
10. Comparing (f ⨉ g)(x) = cos(2x) and h(x) = cos x, we notice that the only difference is the argument of the cosine function.
11. The argument of the cosine function in (f ⨉ g)(x) is 2x, while in h(x), it is x.
12. By comparing these arguments, we can see that the graph of (f ⨉ g)(x) is equivalent to the graph of h(x) horizontally compressed by a factor of 1/2.
13. This compression occurs because for any value of x, the corresponding value of 2x is half as large as x.
14. Therefore, the graph of (f ⨉ g)(x) is equivalent to the graph of h(x)=cos x after being horizontally compressed by a factor of 1/2.
In summary, the product of f(x) and g(x) results in (f ⨉ g)(x) = cos(2x). Comparing this expression to h(x) = cos x, we can observe that (f ⨉ g)(x) is equivalent to h(x) horizontally compressed by a factor of 1/2.