To analyze this problem, let's work through each of the questions based on the given tables for functions $f(x)$, $g(x)$, and $h(x)$.

First Problem Analysis for $g(x)$

We need to express $g(x)$ in terms of $f(x)$.

Comparing $g(x)$ and $f(x)$:

Let's look at the values of $g(x)$ and see how they relate to $f(x)$:

• For $x = -3$, $g(-3) = -2$ and $f(-2) = -2$.
• For $x = -2$, $g(-2) = 0$ and $f(-1) = 0$.
• For $x = -1$, $g(-1) = 3$ and $f(0) = 3$.
• For $x = 0$, $g(0) = -4$ and $f(1) = -4$.
• For $x = 1$, $g(1) = 1$ and $f(2) = 1$.

From this, we observe that $g(x) = f(x + 1)$ because for each $x$ in $g(x)$, the corresponding value of $g(x)$ is equal to $f(x + 1)$.

Thus, the answer for $g(x)$ is: $g(x) = f(x + 1)$

Second Problem Analysis for $h(x)$

We need to express $h(x)$ in terms of $f(x)$.

Comparing $h(x)$ and $f(x)$:

Now let's check how $h(x)$ relates to $f(x)$:

• For $x = -2$, $h(-2) = -1$ and $f(-2) = -2$.
• For $x = -1$, $h(-1) = 1$ and $f(-1) = 0$.
• For $x = 0$, $h(0) = -2$ and $f(0) = -3$.
• For $x = 1$, $h(1) = -3$ and $f(1) = -4$.
• For $x = 2$, $h(2) = 2$ and $f(2) = 1$.

From this comparison, we observe that $h(x) = f(x) + 1$, as each value of $h(x)$ is exactly one unit greater than the corresponding value of $f(x)$.

Thus, the answer for $h(x)$ is: $h(x) = f(x) + 1$

Would you like further details on any of these solutions?

Related Questions:

1. How would you determine $f(x - 1)$ from a given table?
2. How can shifts in functions (like $f(x + 1)$ or $f(x) + 1$) affect their graphs?
3. What would $g(x - 1)$ look like in terms of values if it were added to the table?
4. How do vertical and horizontal shifts of functions differ?
5. Can you provide examples of transformations using different base functions?

Tip:

When analyzing function tables, look for patterns between inputs and outputs to spot transformations like shifts or scaling.