Tables of function values for f(x), g(x), and h(x) at given x-values.

Problem Analysis:

The task involves analyzing three functions, \( f(x) \), \( g(x) \), and \( h(x) \), based on the provided tables of values. The goal is to identify the nature of each function (e.g., linear, quadratic, exponential, etc.) and understand their behavior.

#### Step 1: Analyze \( f(x) \)
The table for \( f(x) \) is as follows:

| \( x \) | \( f(x) \) |
|---------|------------|
| \(-1\) | \( \frac{1}{3} \) |
| \( 0 \) | \( 1 \) |
| \( 1 \) | \( 3 \) |
| \( 2 \) | \( 9 \) |
| \( 3 \) | \( 27 \) |
| \( 4 \) | \( 81 \) |

- Pattern Observation:
- Notice that \( f(x) \) appears to be growing exponentially.
- Specifically, \( f(x) = 3^x \):
- \( f(-1) = 3^{-1} = \frac{1}{3} \)
- \( f(0) = 3^0 = 1 \)
- \( f(1) = 3^1 = 3 \)
- \( f(2) = 3^2 = 9 \)
- \( f(3) = 3^3 = 27 \)
- \( f(4) = 3^4 = 81 \)

- Conclusion: \( f(x) \) is an exponential function with the form \( f(x) = 3^x \).

#### Step 2: Analyze \( g(x) \)
The table for \( g(x) \) is as follows:

| \( x \) | \( g(x) \) |
|---------|------------|
| \(-1\) | \(-8\) |
| \( 0 \) | \(-5\) |
| \( 1 \) | \(-2\) |
| \( 2 \) | \( 1 \) |
| \( 3 \) | \( 4 \) |
| \( 4 \) | \( 7 \) |

- Pattern Observation:
- Calculate the differences between consecutive values of \( g(x) \):
- \( g(0) - g(-1) = -5 - (-8) = 3 \)
- \( g(1) - g(0) = -2 - (-5) = 3 \)
- \( g(2) - g(1) = 1 - (-2) = 3 \)
- \( g(3) - g(2) = 4 - 1 = 3 \)
- \( g(4) - g(3) = 7 - 4 = 3 \)
- The differences are constant (\( \Delta g(x) = 3 \)), indicating a linear function.

- Form of the Linear Function:
- The general form of a linear function is \( g(x) = mx + b \).
- From the table, when \( x = 0 \), \( g(0) = -5 \). Thus, \( b = -5 \).
- The slope \( m \) is the constant difference, which is \( 3 \).
- Therefore, \( g(x) = 3x - 5 \).

- Conclusion: \( g(x) \) is a linear function with the form \( g(x) = 3x - 5 \).

#### Step 3: Analyze \( h(x) \)
The table for \( h(x) \) is as follows:

| \( x \) | \( h(x) \) |
|---------|------------|
| \(-1\) | \( 0 \) |
| \( 0 \) | \(-1\) |
| \( 1 \) | \( 0 \) |
| \( 2 \) | \( 3 \) |
| \( 3 \) | \( 8 \) |
| \( 4 \) | \( 15 \) |

- Pattern Observation:
- Calculate the first differences (\( \Delta h(x) \)):
- \( h(0) - h(-1) = -1 - 0 = -1 \)
- \( h(1) - h(0) = 0 - (-1) = 1 \)
- \( h(2) - h(1) = 3 - 0 = 3 \)
- \( h(3) - h(2) = 8 - 3 = 5 \)
- \( h(4) - h(3) = 15 - 8 = 7 \)
- The first differences are not constant, so \( h(x) \) is not linear.
- Calculate the second differences (\( \Delta^2 h(x) \)):
- \( \Delta h(1) - \Delta h(0) = 1 - (-1) = 2 \)
- \( \Delta h(2) - \Delta h(1) = 3 - 1 = 2 \)
- \( \Delta h(3) - \Delta h(2) = 5 - 3 = 2 \)
- \( \Delta h(4) - \Delta h(3) = 7 - 5 = 2 \)
- The second differences are constant (\( \Delta^2 h(x) = 2 \)), indicating a quadratic function.

- Form of the Quadratic Function:
- The general form of a quadratic function is \( h(x) = ax^2 + bx + c \).
- Use the points from the table to solve for \( a \), \( b \), and \( c \):
- When \( x = 0 \), \( h(0) = -1 \): \( c = -1 \).
- When \( x = 1 \), \( h(1) = 0 \): \( a(1)^2 + b(1) + c = 0 \Rightarrow a + b - 1 = 0 \Rightarrow a + b = 1 \).
- When \( x = 2 \), \( h(2) = 3 \): \( a(2)^2 + b(2) + c = 3 \Rightarrow 4a + 2b - 1 = 3 \Rightarrow 4a + 2b = 4 \Rightarrow 2a + b = 2 \).
- Solve the system of equations:
\[
\begin{cases}
a + b = 1 \\
2a + b = 2
\end{cases}
\]
Subtract the first equation from the second:
\[
(2a + b) - (a + b) = 2 - 1 \Rightarrow a = 1
\]
Substitute \( a = 1 \) into \( a + b = 1 \):
\[
1 + b = 1 \Rightarrow b = 0
\]
- Therefore, \( h(x) = x^2 - 1 \).

- Conclusion: \( h(x) \) is a quadratic function with the form \( h(x) = x^2 - 1 \).

Final Answer:

\[
\boxed{f(x) = 3^x, \, g(x) = 3x - 5, \, h(x) = x^2 - 1}
\]