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) after it has been horizontally compressed by a factor of 1/2, we need to break down the expression and analyze the properties of each function.
Let's start by computing (f * g)(x), which represents the multiplication of f(x) and g(x):
(f * g)(x) = f(x) * g(x) = (cos x - sin x) * (cos x + sin x)
= cos^2(x) - sin^2(x)
Now, let's consider the graph of h(x) = cos x. The cosine function is known for its waveform, oscillating between values -1 and 1 depending on the value of x.
To see the relationship between (f * g)(x) and h(x), let's simplify (f * g)(x) further using trigonometric identities:
cos^2(x) - sin^2(x)
= (cos x + sin x)(cos x - sin x) / (cos x + sin x)
= (cos x + sin x) * 1 / (cos x + sin x)
= 1
From this simplification, we can observe that (f * g)(x) reduces to a constant value of 1, regardless of the value of x.
Now, let's consider the effect of horizontally compressing the graph. When a graph is compressed horizontally by a factor of 1/k, where k is a positive constant, every point (x, f(x)) on the original graph shifts to the right by a factor of k. In other words, the x-values are multiplied by k.
In this case, we want to horizontally compress the graph of h(x) = cos x by a factor of 1/2. Hence, every point on the graph of h(x) will shift to the right by multiplying the x-values by 2.
Since (f * g)(x) simplifies to a constant value of 1, every point on the graph of (f * g)(x) will have the same y-value. Thus, the horizontal compression by a factor of 1/2 will not affect the graph because all points lie on the same horizontal line.
Therefore, we conclude that 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.