To find [tex]\( f(3) \)[/tex] for the function [tex]\( f(x) = -2x^2 + 5 \sqrt{x-2} \)[/tex], we need to substitute [tex]\( x \)[/tex] with 3 in the given function and simplify the expression step-by-step.
1. Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[ f(3) = -2(3)^2 + 5 \sqrt{3-2} \][/tex]
2. Calculate [tex]\( (3)^2 \)[/tex]:
[tex]\[ f(3) = -2(9) + 5 \sqrt{3-2} \][/tex]
3. Calculate [tex]\( -2 \times 9 \)[/tex]:
[tex]\[ f(3) = -18 + 5 \sqrt{1} \][/tex]
4. Simplify the square root:
[tex]\[ f(3) = -18 + 5 \times 1 \][/tex]
5. Multiply [tex]\( 5 \times 1 \)[/tex]:
[tex]\[ f(3) = -18 + 5 \][/tex]
6. Add [tex]\(-18\)[/tex] and [tex]\(5\)[/tex]:
[tex]\[ f(3) = -13 \][/tex]
Thus, the value of [tex]\( f(3) \)[/tex] is
[tex]\[ f(3) = -13 \][/tex]
