Certainly! Let's tackle each part of the problem step-by-step.
### Part (a): Finding [tex]\( g(0) \)[/tex]
To find [tex]\( g(0) \)[/tex], we need to substitute [tex]\( t = 0 \)[/tex] into the function [tex]\( g(t) \)[/tex].
The function given is:
[tex]\[ g(t) = \frac{1}{t + 2} - 1 \][/tex]
So, substituting [tex]\( t = 0 \)[/tex] into the function, we get:
[tex]\[ g(0) = \frac{1}{0 + 2} - 1 \][/tex]
Simplifying inside the fraction:
[tex]\[ g(0) = \frac{1}{2} - 1 \][/tex]
Now, subtract:
[tex]\[ g(0) = \frac{1}{2} - \frac{2}{2} \][/tex]
[tex]\[ g(0) = \frac{1 - 2}{2} \][/tex]
[tex]\[ g(0) = \frac{-1}{2} \][/tex]
[tex]\[ g(0) = -0.5 \][/tex]
Thus,
[tex]\[ g(0) = -0.5 \][/tex]
### Part (b): Solving [tex]\( g(t) = 0 \)[/tex]
To solve [tex]\( g(t) = 0 \)[/tex], we set the function equal to zero and solve for [tex]\( t \)[/tex].
The function is:
[tex]\[ g(t) = \frac{1}{t + 2} - 1 \][/tex]
Setting this equal to zero:
[tex]\[ \frac{1}{t + 2} - 1 = 0 \][/tex]
Add 1 to both sides to isolate the fraction:
[tex]\[ \frac{1}{t + 2} = 1 \][/tex]
Next, we set the fraction equal to 1:
[tex]\[ 1 = t + 2 \][/tex]
To solve for [tex]\( t \)[/tex], subtract 2 from both sides:
[tex]\[ t = 1 - 2 \][/tex]
[tex]\[ t = -1 \][/tex]
Therefore,
[tex]\[ t = -1 \][/tex]
### Summary
- For part (a), [tex]\( g(0) = -0.5 \)[/tex].
- For part (b), the solution to [tex]\( g(t) = 0 \)[/tex] is [tex]\( t = -1 \)[/tex].
So, the answers are:
[tex]\[ g(0) = -0.5 \][/tex]
[tex]\[ t = -1 \][/tex]
