To determine which function $h(x)$ corresponds to from the options provided, let's analyze the tables for $f(x)$ and $h(x)$ to see if there is a consistent relationship between them.

Given Information:

1. Table for $f(x)$:

   • $f(-2) = 4$
   • $f(-1) = 2$
   • $f(0) = 0$
   • $f(1) = 1$
   • $f(2) = -3$

2. Table for $h(x)$:

   • $h(-2) = 3$
   • $h(-1) = 1$
   • $h(0) = -1$
   • $h(1) = 0$
   • $h(2) = -4$

Analyzing the Possible Relationships

Let's test each option by checking if applying the transformation to $f(x)$ yields $h(x)$ values.

Option 1: $h(x) = f(x - 1)$

• $h(-2) = f(-3)$ (not in the table, so skip this)
• This option does not match with the given $x$-values.

Option 2: $h(x) = f(x) + 1$

• $h(-2) = f(-2) + 1 = 4 + 1 = 5$ (does not match $h(-2) = 3$)
• This option does not match either.

Option 3: $h(x) = f(x + 1)$

• $h(-2) = f(-1) = 2$
• $h(-1) = f(0) = 0$
• $h(0) = f(1) = 1$
• $h(1) = f(2) = -3$
• $h(2) = f(3)$ (not in the table)

For the given values of $x$ from -2 to 1, this option matches the table values.

Option 4: $h(x) = f(x) - 1$

• $h(-2) = f(-2) - 1 = 4 - 1 = 3$
• $h(-1) = f(-1) - 1 = 2 - 1 = 1$
• $h(0) = f(0) - 1 = 0 - 1 = -1$
• $h(1) = f(1) - 1 = 1 - 1 = 0$
• $h(2) = f(2) - 1 = -3 - 1 = -4$

This option is consistent with all $h(x)$ values in the table.

Conclusion

The correct answer is: $h(x) = f(x) - 1$

Would you like further explanation on this problem?

Related Questions:

1. How would you find $h(x)$ if a different transformation were applied to $f(x)$?
2. Can you express $h(x)$ in terms of a different shift or scaling of $f(x)$?
3. What other transformations could result in a shift or alteration of ( f(x) \