Question

Select the best description for each of the following functions defined form real number to real numbers.

g(x) = x + 1 − 3x
1. injective function
2. surjective function
f(x) = [x²] - x
3. bijective function.
4. non-injective and non- surjective function

Answer 1

g(x) = -2x + 1 is bijective, as it is a linear function with a non-zero slope, hence both injective and surjective. The function f(x) = [x²] - x might not be injective or surjective if the brackets denote the floor function, but it's hard to determine without additional context.

Step-by-step explanation:

To answer the student's question, let's analyze each given function separately:

• g(x) = x + 1 − 3x simplifies to g(x) = -2x + 1. As a linear function with a non-zero slope, it is both injective (one-to-one) and surjective (onto) since for every value of y in the codomain, there is exactly one value of x in the domain that maps to it. Thus, this function is bijective as it is both injective and surjective.
• The function f(x) = [x²] - x is a bit more complex because of the square brackets, which usually denote the floor function or integer part of x². Without more context it's hard to provide a definitive answer, but if we assume the brackets mean the floor function, then this function is not injective, as multiple values of x can yield the same output once the floor is taken. It's hard to determine surjectivity without more context about the codomain, but typically, such floor functions are not surjective on the set of all real numbers because there will be real numbers that cannot be achieved exactly by the function. Therefore, in this case, we could say it's a non-injective and non-surjective function.