Final answer:
The function f(x) = x² + 1 is not one-to-one because it does not pass the horizontal line test; a horizontal line intersects the graph of the function in more than one point for y-values greater than 1.
Step-by-step explanation:
The question is asking whether the function f(x) = x² + 1 is one-to-one for all real numbers except when x ≠ 1. To determine this, we need to understand what a one-to-one function is. A function is one-to-one if, and only if, every y-value in the range is matched with exactly one x-value in the domain. Another way to look at it is that no horizontal line intersects the graph of the function at more than one point. For the quadratic function given, f(x) = x² + 1, a horizontal line will intersect the graph in two places for any y-value greater than 1. Therefore, the function is not one-to-one. For example, f(-2) = 2² + 1 = 5 and f(2) = 2² + 1 = 5; both -2 and 2 give the same output, illustrating that it is not one-to-one.
The correct answer is No, because the function is not one-to-one; however, the reason provided in the question's options does not fully capture why. The proper justification is because the function does not pass the horizontal line test.