Let's complete the table step-by-step, considering [tex]\( f \)[/tex] and [tex]\( g \)[/tex] as one-to-one functions.
### Given Data
- Values of [tex]\( f(x) \)[/tex]: [tex]\( f(-9) = -8 \)[/tex], [tex]\( f(-8) = 15 \)[/tex], and [tex]\( f(7) \)[/tex] and [tex]\( f(15) \)[/tex] are not defined.
- Values of [tex]\( g(x) \)[/tex]: [tex]\( g(-9) = 7 \)[/tex], [tex]\( g(-8) \)[/tex] is not defined, [tex]\( g(7) = 15 \)[/tex], and [tex]\( g(15) = 7 \)[/tex].
### Step-by-Step Solution
1. Complete [tex]\( f(x) \)[/tex]:
- Given [tex]\( f(-9) = -8 \)[/tex] and [tex]\( f(-8) = 15 \)[/tex].
- But [tex]\( f(7) \)[/tex] and [tex]\( f(15) \)[/tex] are not given, implying [tex]\( f(7) \)[/tex] and [tex]\( f(15) \)[/tex] are undefined.
2. Complete [tex]\( g(x) \)[/tex]:
- Given [tex]\( g(-9) = 7 \)[/tex], [tex]\( g(7) = 15 \)[/tex], and [tex]\( g(15) = 7 \)[/tex].
- [tex]\( g(-8) \)[/tex] is not given, implying [tex]\( g(-8) \)[/tex] is undefined.
3. Calculate [tex]\((f \circ g)(x)\)[/tex]:
[tex]\((f \circ g)(x)\)[/tex] means applying [tex]\( g \)[/tex] first and then [tex]\( f \)[/tex].
- [tex]\((f \circ g)(-9) = f(g(-9)) = f(7)\)[/tex]. Since [tex]\( f(7) \)[/tex] is not defined, [tex]\((f \circ g)(-9)\)[/tex] is undefined.
- [tex]\((f \circ g)(-8) = f(g(-8))\)[/tex]. Since [tex]\( g(-8) \)[/tex] is not defined, [tex]\((f \circ g)(-8)\)[/tex] is undefined.
- [tex]\((f \circ g)(7) = f(g(7)) = f(15)\)[/tex]. Since [tex]\( f(15) \)[/tex] is not defined, [tex]\((f \circ g)(7)\)[/tex] is undefined.
- [tex]\((f \circ g)(15) = f(g(15)) = f(7)\)[/tex]. Since [tex]\( f(7) \)[/tex] is not defined, [tex]\((f \circ g)(15)\)[/tex] is undefined.
4. Calculate [tex]\((g \circ f)(x)\)[/tex]:
[tex]\((g \circ f)(x)\)[/tex] means applying [tex]\( f \)[/tex] first and then [tex]\( g \)[/tex].
- [tex]\((g \circ f)(-9) = g(f(-9)) = g(-8)\)[/tex]. Since [tex]\( g(-8) \)[/tex] is not defined, [tex]\((g \circ f)(-9)\)[/tex] is undefined.
- [tex]\((g \circ f)(-8) = g(f(-8)) = g(15) = 7\)[/tex].
- [tex]\((g \circ f)(7)\)[/tex] is not computable since [tex]\( f(7) \)[/tex] is not defined.
- [tex]\((g \circ f)(15)\)[/tex] is not computable since [tex]\( f(15) \)[/tex] is not defined.
### Final Table
[tex]\[
\begin{array}{|l|l|l|l|l|}
\hline
x & -9 & -8 & 7 & 15 \\
\hline
f(x) & -8 & 15 & \text{undefined} & \text{undefined} \\
\hline
g(x) & 7 & \text{undefined} & 15 & 7 \\
\hline
(f \circ g)(x) & \text{undefined} & \text{undefined} & \text{undefined} & \text{undefined} \\
\hline
(g \circ f)(x) & \text{undefined} & 7 & \text{undefined} & \text{undefined} \\
\hline
\end{array}
\][/tex]
