Final answer:
The first function, f(x) = 2x + 1, is a function from R to R. The second function, f(x) = x² + 1, is a function from R to R but it is not a bijection.
Step-by-step explanation:
The first function, f(x) = 2x + 1, is a function from R to R because it takes a real number input x and produces a real number output. To check if it is a function, we need to make sure that for every x in R, there is a unique y in R. In this case, for every x value, we can calculate a unique y value using the equation 2x + 1. Therefore, this function is a function from R to R.
The second function, f(x) = x² + 1, is also a function from R to R. However, it is not a bijection. A bijection is a function that is both injective (one-to-one) and surjective (onto).
- An injective function means that each element in the domain has a unique corresponding element in the codomain. In other words, no two different x values can have the same y value. In the case of f(x) = x² + 1, if we take x = 1 and x = -1, both of them will give us the same y value of 2. This violates the injective property, so it is not injective.
- A surjective function means that every element in the codomain has a corresponding element in the domain. In other words, every y value in the codomain can be obtained by plugging in some x value. In the case of f(x) = x² + 1, the function does not cover all real numbers. For example, there is no x value that will give us a y value of -1. Therefore, it is not surjective.
Since f(x) = x² + 1 fails to satisfy both injectiveness and surjectiveness, it is not a bijection.