Consider the function f: R→R defined by f(x)=sin(x)+cos(2x). Which of the following statements about f(x) is true? a. The function f(x) is one-to-one (injective). b. The function f(x) is onto (surjective). c. The function f(x) is both one-to-one and onto. d. The function f(x) is neither one-to-one nor onto.

The function f(x) = sin(x) + cos(2x) is neither one-to-one (injective) nor onto (surjective).

Here's why:

a. The function is not one-to-one (injective): A function is said to be injective if every element of the function's domain maps to a unique element of its range. In other words, no two different inputs will produce the same output. However, for the function f(x) = sin(x) + cos(2x), different values of x can produce the same output. For example, sin(0) + cos(20) = sin(pi) + cos(2pi) = 1. Therefore, the function is not injective.

b. The function is not onto (surjective): A function is said to be surjective if every element of the function's range is the image of at least one element of its domain. In other words, for any given output, there is at least one input that will produce it. However, for the function f(x) = sin(x) + cos(2x), the range is not all real numbers. The range of sin(x) is [-1,1] and the range of cos(2x) is also [-1,1]. Therefore, the range of f(x) = sin(x) + cos(2x) is [-2,2], which is not all real numbers. Therefore, the function is not surjective.

So, the correct answer is d. The function f(x) = sin(x) + cos(2x) is neither one-to-one nor onto.