Final answer:
Option (D) f(x) = 2x + 3 is the one-to-one function because it is a linear function with a non-zero slope, ensuring that each x-value maps to a unique y-value.
Step-by-step explanation:
To identify a one-to-one function among the given options, we need to determine which function has the property that each x-value corresponds to one and only one y-value, and each y-value corresponds to one and only one x-value. This is also known as an injective function.
- (A) f(x) = \( \frac{1}{x^2 + 1} \) is not one-to-one because the function is symmetric about the y-axis and thus different x-values will yield the same y-value.
- (B) f(x) = \( \frac{1}{3x^2 + 7x} \) also is not one-to-one due to the same reasons as (A); it fails the horizontal line test.
- (C) f(x) = x^2 is not one-to-one since different x-values (for instance, 1 and -1) will give the same y-value.
- (D) f(x) = 2x + 3 is a linear function with a non-zero slope. This function is indeed one-to-one because, for every x-value, there is a unique y-value, and it passes the horizontal line test.
Therefore, the one-to-one function is option (D) f(x) = 2x + 3.