Sure! Let's go through the evaluation of the function [tex]\( g(x) = -2x^2 + 3x - 5 \)[/tex] for the given input values [tex]\( x = -2, 0, \)[/tex] and [tex]\( 3 \)[/tex].
### Step 1: Evaluate [tex]\( g(-2) \)[/tex]
Given:
[tex]\[ g(x) = -2x^2 + 3x - 5 \][/tex]
Evaluating it at [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -2(-2)^2 + 3(-2) - 5 \][/tex]
Calculate each term:
[tex]\[ (-2)^2 = 4 \][/tex]
[tex]\[ -2 \cdot 4 = -8 \][/tex]
[tex]\[ 3 \cdot -2 = -6 \][/tex]
Putting it all together:
[tex]\[ g(-2) = -8 - 6 - 5 \][/tex]
[tex]\[ g(-2) = -19 \][/tex]
So,
[tex]\[ g(-2) = -19 \][/tex]
### Step 2: Evaluate [tex]\( g(0) \)[/tex]
Evaluating it at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -2(0)^2 + 3(0) - 5 \][/tex]
Calculate each term:
[tex]\[ 0^2 = 0 \][/tex]
[tex]\[ -2 \cdot 0 = 0 \][/tex]
[tex]\[ 3 \cdot 0 = 0 \][/tex]
Putting it all together:
[tex]\[ g(0) = 0 + 0 - 5 \][/tex]
[tex]\[ g(0) = -5 \][/tex]
So,
[tex]\[ g(0) = -5 \][/tex]
### Step 3: Evaluate [tex]\( g(3) \)[/tex]
Evaluating it at [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = -2(3)^2 + 3(3) - 5 \][/tex]
Calculate each term:
[tex]\[ (3)^2 = 9 \][/tex]
[tex]\[ -2 \cdot 9 = -18 \][/tex]
[tex]\[ 3 \cdot 3 = 9 \][/tex]
Putting it all together:
[tex]\[ g(3) = -18 + 9 - 5 \][/tex]
[tex]\[ g(3) = -14 \][/tex]
So,
[tex]\[ g(3) = -14 \][/tex]
### Summary of Results
- [tex]\( g(-2) = -19 \)[/tex]
- [tex]\( g(0) = -5 \)[/tex]
- [tex]\( g(3) = -14 \)[/tex]
These are the evaluations of the function [tex]\( g(x) \)[/tex] for the given input values.
