Let's solve each part of the given function step by step:
The function given is:
[tex]\[ g(t) = 7t^2 - 5t + 5 \][/tex]
### (a) [tex]\( g(2) \)[/tex]
To find [tex]\( g(2) \)[/tex], we substitute [tex]\( t = 2 \)[/tex] into the function.
[tex]\[ g(2) = 7(2)^2 - 5(2) + 5 \][/tex]
Simplify the expression within the function:
[tex]\[ g(2) = 7 \cdot 4 - 5 \cdot 2 + 5 \][/tex]
Now, calculate each term:
[tex]\[ g(2) = 28 - 10 + 5 \][/tex]
Combine these values:
[tex]\[ g(2) = 23 \][/tex]
So, [tex]\( g(2) = 23 \)[/tex].
### (b) [tex]\( g(-1) \)[/tex]
To find [tex]\( g(-1) \)[/tex], we substitute [tex]\( t = -1 \)[/tex] into the function.
[tex]\[ g(-1) = 7(-1)^2 - 5(-1) + 5 \][/tex]
Simplify the expression within the function:
[tex]\[ g(-1) = 7 \cdot 1 + 5 \cdot 1 + 5 \][/tex]
Now, calculate each term:
[tex]\[ g(-1) = 7 + 5 + 5 \][/tex]
Combine these values:
[tex]\[ g(-1) = 17 \][/tex]
So, [tex]\( g(-1) = 17 \)[/tex].
### (c) [tex]\( g(t+2) \)[/tex]
To find [tex]\( g(t+2) \)[/tex], we substitute [tex]\( t = t+2 \)[/tex] into the function.
[tex]\[ g(t+2) = 7(t+2)^2 - 5(t+2) + 5 \][/tex]
First, expand [tex]\( (t+2)^2 \)[/tex]:
[tex]\[ (t+2)^2 = t^2 + 4t + 4 \][/tex]
Substitute this back into the function:
[tex]\[ g(t+2) = 7(t^2 + 4t + 4) - 5(t+2) + 5 \][/tex]
Next, distribute the constants through the parentheses:
[tex]\[ g(t+2) = 7t^2 + 28t + 28 - 5t - 10 + 5 \][/tex]
Combine like terms:
[tex]\[ g(t+2) = 7t^2 + (28t - 5t) + (28 - 10 + 5) \][/tex]
[tex]\[ g(t+2) = 7t^2 + 23t + 23 \][/tex]
So, [tex]\( g(t+2) = 7t^2 + 23t + 23 \)[/tex].
Therefore, the final results are:
- [tex]\( g(2) = 23 \)[/tex]
- [tex]\( g(-1) = 17 \)[/tex]
- [tex]\( g(t+2) = 7t^2 + 23t + 23 \)[/tex]
