Let's evaluate the function [tex]\( g(x) = -2x^2 + 3x - 5 \)[/tex] for the input values [tex]\(-2\)[/tex], [tex]\(0\)[/tex], and [tex]\(3\)[/tex].
### Step-by-Step Solutions:
#### 1. Evaluating [tex]\( g(-2) \)[/tex]:
[tex]\[
g(-2) = -2(-2)^2 + 3(-2) - 5
\][/tex]
First, calculate [tex]\((-2)^2\)[/tex]:
[tex]\[
(-2)^2 = 4
\][/tex]
Now multiply by [tex]\(-2\)[/tex]:
[tex]\[
-2 \cdot 4 = -8
\][/tex]
Next, calculate [tex]\(3 \cdot (-2)\)[/tex]:
[tex]\[
3 \cdot (-2) = -6
\][/tex]
Finally, sum all terms:
[tex]\[
g(-2) = -8 - 6 - 5 = -19
\][/tex]
So,
[tex]\[
g(-2) = -19
\][/tex]
#### 2. Evaluating [tex]\( g(0) \)[/tex]:
[tex]\[
g(0) = -2(0)^2 + 3(0) - 5
\][/tex]
First, calculate [tex]\(0^2\)[/tex]:
[tex]\[
0^2 = 0
\][/tex]
Now multiply by [tex]\(-2\)[/tex]:
[tex]\[
-2 \cdot 0 = 0
\][/tex]
Next, calculate [tex]\(3 \cdot 0\)[/tex]:
[tex]\[
3 \cdot 0 = 0
\][/tex]
Finally, sum all terms:
[tex]\[
g(0) = 0 + 0 - 5 = -5
\][/tex]
So,
[tex]\[
g(0) = -5
\][/tex]
#### 3. Evaluating [tex]\( g(3) \)[/tex]:
[tex]\[
g(3) = -2(3)^2 + 3(3) - 5
\][/tex]
First, calculate [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
Now multiply by [tex]\(-2\)[/tex]:
[tex]\[
-2 \cdot 9 = -18
\][/tex]
Next, calculate [tex]\(3 \cdot 3\)[/tex]:
[tex]\[
3 \cdot 3 = 9
\][/tex]
Finally, sum all terms:
[tex]\[
g(3) = -18 + 9 - 5 = -14
\][/tex]
So,
[tex]\[
g(3) = -14
\][/tex]
### Final Results:
[tex]\[
\begin{array}{l}
g(-2) = -19 \\
g(0) = -5 \\
g(3) = -14 \\
\end{array}
\][/tex]
