Questions: A function is given by a formula. Determine whether it is one-to-one. g(x)=cos(x) Yes, it is one-to-one. No, it is not one-to-one.

Solution Steps

To determine if a function is one-to-one, we need to check if different inputs produce different outputs. For the cosine function, we know that it is periodic and repeats its values over intervals. Therefore, it is not one-to-one.

The function given is \( g(x) = \cos(x) \). The cosine function is a trigonometric function that is periodic with a period of \( 2\pi \). This means that the function repeats its values every \( 2\pi \) units.

A function is one-to-one if and only if different inputs produce different outputs. For the cosine function, we can check if there exist different values of \( x \) that produce the same output.

Let's evaluate the function at two different points: \( x_1 = 0 \) and \( x_2 = 2\pi \).

\[ g(0) = \cos(0) = 1 \]

\[ g(2\pi) = \cos(2\pi) = 1 \]

Since \( g(0) = g(2\pi) \), the function produces the same output for different inputs.

Since the function \( g(x) = \cos(x) \) produces the same output for different inputs, it is not one-to-one.

Final Answer