To determine [tex]\( |f(i)| \)[/tex] for the function [tex]\( f(x) = 1 - x \)[/tex], follow these steps:
1. Substitute the imaginary unit [tex]\( i \)[/tex] into the function:
[tex]\[
f(i) = 1 - i
\][/tex]
2. Represent [tex]\( f(i) \)[/tex] as a complex number:
[tex]\[
f(i) = 1 - i = 1 + (-i)
\][/tex]
Here, [tex]\( 1 \)[/tex] is the real part and [tex]\( -i \)[/tex] is the imaginary part.
3. Calculate the magnitude (or absolute value) of the complex number [tex]\( 1 - i \)[/tex]:
The magnitude [tex]\( |a + bi| \)[/tex] of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[
|a + bi| = \sqrt{a^2 + b^2}
\][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -1 \)[/tex], so:
[tex]\[
|1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\][/tex]
Thus, the value equivalent to [tex]\( |f(i)| \)[/tex] is [tex]\(\sqrt{2}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\sqrt{2}
\][/tex]
