To determine whether the function [tex]\( f(x) = \sqrt{19 - x^2} \)[/tex] is one-to-one, we need to check if each value of [tex]\( f(x) \)[/tex] corresponds to only one value of [tex]\( x \)[/tex].
For a function to be one-to-one, different values of [tex]\( x \)[/tex] must yield different values of [tex]\( f(x) \)[/tex]. In other words, if [tex]\( x_1 \neq x_2 \)[/tex], then [tex]\( f(x_1) \neq f(x_2) \)[/tex].
Let's evaluate the function at two different [tex]\( x \)[/tex]-values: [tex]\( x_1 = 2 \)[/tex] and [tex]\( x_2 = -2 \)[/tex].
1. Compute [tex]\( f(2) \)[/tex]:
[tex]$ f(2) = \sqrt{19 - (2)^2} = \sqrt{19 - 4} = \sqrt{15} $[/tex]
2. Compute [tex]\( f(-2) \)[/tex]:
[tex]$ f(-2) = \sqrt{19 - (-2)^2} = \sqrt{19 - 4} = \sqrt{15} $[/tex]
We see that:
[tex]$ f(2) = \sqrt{15} $[/tex]
[tex]$ f(-2) = \sqrt{15} $[/tex]
Since [tex]\( f(2) = f(-2) \)[/tex], we have found two different [tex]\( x \)[/tex]-values (namely [tex]\( 2 \)[/tex] and [tex]\( -2 \)[/tex]) that correspond to the same [tex]\( f(x) \)[/tex] value ([tex]\( \sqrt{15} \)[/tex]).
Therefore, the function [tex]\( f(x) = \sqrt{19 - x^2} \)[/tex] is not one-to-one because there exist at least two distinct [tex]\( x \)[/tex]-values that produce the same [tex]\( f(x) \)[/tex] value.
The correct choice is:
A. No, because the two [tex]\( x \)[/tex]-values 2 and -2 correspond to the same [tex]\( f(x) \)[/tex] value [tex]\( \sqrt{15} \)[/tex].
