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Prove that [tex]\( g \)[/tex] is not one-to-one using the definition of a one-to-one function.

[tex]\[ g(x) = 4 - x^2 \][/tex]

Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

A. There exist two numbers [tex]\( 1 \)[/tex] and [tex]\( -1 \)[/tex] such that [tex]\( 1 \neq -1 \)[/tex], and [tex]\( g(1) = g(-1) = (0, 4) \)[/tex]. Thus, [tex]\( g \)[/tex] is not one-to-one.
B. There exist two numbers [tex]\( 1 \)[/tex] and [tex]\( -1 \)[/tex] such that [tex]\( 1 \neq -1 \)[/tex], and [tex]\( g(1) = \square \)[/tex] and [tex]\( g(-1) = \square \)[/tex]. Thus, [tex]\( g \)[/tex] is not one-to-one.



Answer :

GinnyAnswer
To prove that the function [tex]\( g(x) = 4 - x^2 \)[/tex] is not one-to-one, we need to show that there exist distinct values [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] such that [tex]\( g(x_1) = g(x_2) \)[/tex] while [tex]\( x_1 \neq x_2 \)[/tex].

Let's begin by selecting two distinct numbers:
[tex]\[ x_1 = 1 \][/tex]
[tex]\[ x_2 = -1 \][/tex]

Next, we evaluate the function [tex]\( g(x) \)[/tex] at these points.

1. Calculate [tex]\( g(1) \)[/tex]:
[tex]\[ g(1) = 4 - (1)^2 = 4 - 1 = 3 \][/tex]

2. Calculate [tex]\( g(-1) \)[/tex]:
[tex]\[ g(-1) = 4 - (-1)^2 = 4 - 1 = 3 \][/tex]

As we can see, even though [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are different:
[tex]\[ 1 \neq -1 \][/tex]

The function values are equal:
[tex]\[ g(1) = g(-1) = 3 \][/tex]

Since there exist at least two distinct inputs (1 and -1) that produce the same output (3), we conclude that the function [tex]\( g(x) = 4 - x^2 \)[/tex] is not one-to-one.

Therefore, the correct choice is:
A. There exist two numbers [tex]\( 1 \)[/tex] and [tex]\( -1 \)[/tex] such that [tex]\( 1 \neq -1 \)[/tex], and [tex]\( g(1) = g(-1) = 3 \)[/tex]. Thus, [tex]\( g \)[/tex] is not one-to-one.

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