Question

Which statement accurately describes the difference between g(x) = cos^2(x) and f(x) = cos(x^2)?

A) g(x) and f(x) are the same; there are no differences between them.
B) g(x) squares the cosine of x first and then takes the input value x, whereas f(x) squares x first and then takes x squared as the input.
C) g(x) takes x as an input, then squares the output, cos(x), whereas f(x) squares x first and then takes x squared as input.
D) g(x) is cos(x) cos(x), whereas f(x) is cos(x) multiplied by x

Answer 1

The difference between g(x) = cos^2(x) and f(x) = cos(x^2) lies in the order of operations: g(x) involves squaring the cosine of x, and f(x) involves taking the cosine of x².

Step-by-step explanation:

The correct answer to the question is C) g(x) takes x as an input, then squares the output, cos(x), whereas f(x) squares x first and then takes x squared as input. This difference signifies that in the function g(x) = cos2(x), we first calculate the cosine of x and then square the resulting value. In contrast, for the function f(x) = cos(x2), we first square the input x and then take the cosine of this squared value. The two functions will yield different outcomes for the same input value x because they process the input and apply the trigonometric cosine function at different stages.