Question

Suppose the function f(x) = cos(x) is transformed and the new rule for the translated function, g(x), is g(x) = cos(3x). Describe this transformation.

Answer 1

The translation of the function f(x) = cos(x) to g(x) = cos(3x) results in a horizontal compression of the cosine function, increasing the frequency and reducing the period of the original function by a factor of 3.

Step-by-step explanation:

When the function f(x) = cos(x) is transformed into g(x) = cos(3x), this represents a horizontal compression of the original cosine function. In the context of transformations, this effect is known as a change in the frequency of the function, where the factor inside the cosine function alters the rate at which the function completes its cycles.

In this case, the factor is 3, which means that the cosine function now completes 3 cycles in the interval where it originally completed just 1 cycle.

Therefore, the period of the cosine function has been reduced by a factor of 3, and the new period is π/3 because the period of the original cos(x) function is 2π and the new period is 2π/3.