To solve the given problem, we need to find [tex]\((g + h)(1)\)[/tex]. We'll follow a detailed step-by-step approach.
Given functions:
[tex]\[ g(t) = 4t^2 - 4t \][/tex]
[tex]\[ h(t) = 2t^2 - 5t + 1 \][/tex]
1. Evaluate [tex]\( g(t) \)[/tex] at [tex]\( t = 1 \)[/tex]:
[tex]\[
g(1) = 4(1)^2 - 4(1)
\][/tex]
Simplify the expression:
[tex]\[
g(1) = 4 \cdot 1 - 4 \cdot 1
\][/tex]
[tex]\[
g(1) = 4 - 4 = 0
\][/tex]
2. Evaluate [tex]\( h(t) \)[/tex] at [tex]\( t = 1 \)[/tex]:
[tex]\[
h(1) = 2(1)^2 - 5(1) + 1
\][/tex]
Simplify the expression:
[tex]\[
h(1) = 2 \cdot 1^2 - 5 \cdot 1 + 1
\][/tex]
[tex]\[
h(1) = 2 - 5 + 1
\][/tex]
[tex]\[
h(1) = -2
\][/tex]
3. Find [tex]\((g + h)(1)\)[/tex]:
[tex]\[
(g + h)(1) = g(1) + h(1)
\][/tex]
Substitute the values obtained:
[tex]\[
(g + h)(1) = 0 + (-2)
\][/tex]
[tex]\[
(g + h)(1) = -2
\][/tex]
Thus, the step-by-step solution yields:
[tex]\[
g(1) = 0, \quad h(1) = -2, \quad \text{and} \quad (g + h)(1) = -2
\][/tex]
