To find [tex]\( f(1) \)[/tex] for the function [tex]\( f(x) = \sqrt{x-10} \)[/tex]:
1. Substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[
f(1) = \sqrt{1 - 10}
\][/tex]
2. Calculate the expression inside the square root:
[tex]\[
1 - 10 = -9
\][/tex]
3. Take the square root of -9. This involves considering complex numbers since the square root of a negative number is not defined in the set of real numbers:
[tex]\[
\sqrt{-9} = \sqrt{(-1) \cdot 9} = \sqrt{-1} \cdot \sqrt{9}
\][/tex]
4. Simplify each square root. Recall that [tex]\( \sqrt{-1} \)[/tex] is denoted as [tex]\( i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit:
[tex]\[
\sqrt{9} = 3 \quad \text{and} \quad \sqrt{-1} = i
\][/tex]
Combining these, we get:
[tex]\[
\sqrt{-9} = 3i
\][/tex]
Thus, [tex]\( f(1) = 3i \)[/tex].
The result can be expressed in a standard complex form, where the real part is zero and the imaginary part is [tex]\( 3 \)[/tex]:
[tex]\[
f(1) = (0 + 3i) \quad \text{or more compactly} \quad (1.8369701987210297e-16 + 3j)
\][/tex]
Therefore,
[tex]\[
f(1) = (1.8369701987210297e-16 + 3j)
\][/tex]
