To determine whether the function [tex]\( f(x) = \sqrt{24 - x^2} \)[/tex] is one-to-one, we need to check whether each [tex]\( y \)[/tex]-value (output) corresponds to exactly one [tex]\( x \)[/tex]-value (input).
1. Evaluate the function at [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = \sqrt{24 - 2^2} = \sqrt{24 - 4} = \sqrt{20} = 2 \sqrt{5}
\][/tex]
2. Evaluate the function at [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = \sqrt{24 - (-2)^2} = \sqrt{24 - 4} = \sqrt{20} = 2 \sqrt{5}
\][/tex]
3. Compare the results:
We see that:
[tex]\[
f(2) = 2 \sqrt{5}
\][/tex]
and
[tex]\[
f(-2) = 2 \sqrt{5}
\][/tex]
Thus, the two [tex]\( x \)[/tex]-values [tex]\( 2 \)[/tex] and [tex]\( -2 \)[/tex] correspond to the same [tex]\( f(x) \)[/tex]-value [tex]\( 2 \sqrt{5} \)[/tex].
4. Conclusion:
Since there are two different [tex]\( x \)[/tex]-values that map to the same [tex]\( y \)[/tex]-value, the function is not one-to-one.
Therefore, the correct choice is:
[tex]\[ \text{A. No, because the two } x \text{-values 2 and -2 correspond to the same } f(x) \text{ value } 2\sqrt{5}. \][/tex]
The values in the answer box should be:
[tex]\[ 2\sqrt{5} \quad \text{and again} \quad 2\sqrt{5}. \][/tex]
