To evaluate the given expressions using the provided table, let's break down each step in detail.
### Step-by-step evaluations:
1. Evaluate [tex]\( f(g(8)) \)[/tex]
- First, find [tex]\( g(8) \)[/tex]:
[tex]\[ g(8) = 3 \][/tex]
- Next, use this result to find [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = 3 \][/tex]
- Therefore,
[tex]\[ f(g(8)) = 3 \][/tex]
2. Evaluate [tex]\( g(f(0)) \)[/tex]
- First, find [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 6 \][/tex]
- Next, use this result to find [tex]\( g(6) \)[/tex]:
[tex]\[ g(6) = 6 \][/tex]
- Therefore,
[tex]\[ g(f(0)) = 6 \][/tex]
3. Evaluate [tex]\( f(f(9)) \)[/tex]
- First, find [tex]\( f(9) \)[/tex]:
[tex]\[ f(9) = 5 \][/tex]
- Next, use this result to find [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 7 \][/tex]
- Therefore,
[tex]\[ f(f(9)) = 7 \][/tex]
4. Evaluate [tex]\( g(g(7)) \)[/tex]
- First, find [tex]\( g(7) \)[/tex]:
[tex]\[ g(7) = 0 \][/tex]
- Next, use this result to find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 1 \][/tex]
- Therefore,
[tex]\[ g(g(7)) = 1 \][/tex]
### Final Results:
- [tex]\( f(g(8)) = 3 \)[/tex]
- [tex]\( g(f(0)) = 6 \)[/tex]
- [tex]\( f(f(9)) = 7 \)[/tex]
- [tex]\( g(g(7)) = 1 \)[/tex]
These are the solutions to the given expressions.
