Question

Suppose that f is one-to-one. We define the function g as g(x) = f(x^2). Which of the following best characterizes g?

Group of Answer Choices:
A. g is one-to-one.
B. We can't tell whether g is one-to-one or not with the given information.
C. g is not one-to-one.

Answer 1

The function g(x) = f(x^2) is not one-to-one (option C) because for any nonzero x, g(x) = g(-x) since the square of a positive number and its negative are equal, thus yielding the same result for two different inputs.

Step-by-step explanation:

To determine whether the function g(x) = f(x2) is one-to-one given that f is one-to-one, we must examine the properties of the functions involved. A function is one-to-one if and only if no two different inputs produce the same output. Since f is one-to-one, we know that for every unique input x, there is a unique output f(x).

However, when considering g, we note that squaring an input value x results in the same value for x2 as for -x2, because the square of a positive number is equal to the square of its negative counterpart. Therefore, for any nonzero x, g(x) = g(-x) because f(x2) = f((-x)2). This means that g cannot be one-to-one since two different inputs (x and -x) yield the same output.

In conclusion, the function g is not one-to-one because it will yield the same result for both x and -x when x is nonzero.