Question

Based on the following tables, which of the following options is a true description of the graphs of f(x) and g(x)
x f(x)
-2 -6
0 -4
2 -2
4 0
x g(x)
-2 6
0 4
2 2
4 0
Option 1 one graph is a reflection of the other over the x-axis
Option 2 one graph is a reflection of the other over the y-axis
Option 3. One graph is a reflection of the other about the origin

Answer 1

To analyze the given functions \( f(x) \) and \( g(x) \), we will examine the values in the two provided tables.

**Values for \( f(x) \)**:
- \( f(-2) = -6 \)
- \( f(0) = -4 \)
- \( f(2) = -2 \)
- \( f(4) = 0 \)

**Values for \( g(x) \)**:
- \( g(-2) = 6 \)
- \( g(0) = 4 \)
- \( g(2) = 2 \)
- \( g(4) = 0 \)

Next, let's check for the relationships implied by the options:

### Option 1: One graph is a reflection of the other over the x-axis.
If \( g(x) \) is the reflection of \( f(x) \) over the x-axis, we would have:
\[ g(x) = -f(x) \]

Checking this for the given values:
- For \( x = -2 \): \( g(-2) = 6 \) and \( -f(-2) = -(-6) = 6 \) (True)
- For \( x = 0 \): \( g(0) = 4 \) and \( -f(0) = -(-4) = 4 \) (True)
- For \( x = 2 \): \( g(2) = 2 \) and \( -f(2) = -(-2) = 2 \) (True)
- For \( x = 4 \): \( g(4) = 0 \) and \( -f(4) = -(0) = 0 \) (True)

Since all these equalities hold, we conclude **this option is true**.

### Option 2: One graph is a reflection of the other over the y-axis.
If \( g(x) \) is the reflection of \( f(x) \) over the y-axis, we would have:
\[ g(-x) = f(x) \]

We would need to check:
- \( g(-2) = f(2) \rightarrow 6 \neq -2 \) (False)
- \( g(0) = f(0) \rightarrow 4 \neq -4 \) (False)
- \( g(2) = f(-2) \rightarrow 2 \neq -6 \) (False)
- \( g(4) = f(-4) \) (not applicable since \( f(-4) \) is not defined above)

None of these hold. **This option is false**.

### Option 3: One graph is a reflection of the other about the origin.
If \( g(x) \) is the reflection of \( f(x) \) about the origin, we would have:
\[ g(-x) = -f(x) \]

Checking this:
- \( g(-2) = 6 \) and \( -f(2) = -(-2) = 2 \) (False)
- \( g(0) = 4 \) and \( -f(0) = -(-4) = 4 \) (True, but not relevant since others are false)
- \( g(2) = 2 \) and \( -f(-2) = -(-6) = 6 \) (False)
- \( g(4) = 0 \) and \( -f(-4) \) (not applicable)

**This option is false**.

### Conclusion:
The only true description of the graphs of \( f(x) \) and \( g(x) \) is:
**Option 1: One graph is a reflection of the other over the x-axis.**