We are given the graph of a function $g$. We need to: 1) Determine the domain $D_g$ of the function $g$. 2) Compare $g(-1)$ and $g(1)$, then compare $g(-3)$ and $g(3)$. 3) Deduce the relationship between $g(-x)$ and $g(x)$ for all $x \in D_g$. 4) Determine the geometric property verified by the curve $(C_g)$.

1. Problem Description

We are given the graph of a function $g$. We need to:

1) Determine the domain $D_g$ of the function $g$.

2) Compare $g(-1)$ and $g(1)$, then compare $g(-3)$ and $g(3)$.

3) Deduce the relationship between $g(-x)$ and $g(x)$ for all $x \in D_g$.

4) Determine the geometric property verified by the curve $(C_g)$.

2. Solution Steps

1) Determine the domain $D_g$:

From the graph, the function is defined from $x = -3$ to $x = 3$. Therefore, the domain is $D_g = [-3, 3]$.

2) Compare $g(-1)$ and $g(1)$, then compare $g(-3)$ and $g(3)$:

From the graph, $g(-1) \approx -2$ and $g(1) \approx 2$.

Also, $g(-3) = 0$ and $g(3) = 0$.

3) Deduce the relationship between $g(-x)$ and $g(x)$ for all $x \in D_g$:

Observing the values from the previous step: $g(-1) = -g(1)$ and $g(-3) = -g(3) = g(3)$. This suggests that $g(-x) = -g(x)$. Therefore, $g$ is an odd function.

4) Determine the geometric property verified by the curve $(C_g)$:

Since $g(-x) = -g(x)$, the function $g$ is odd. The graph of an odd function is symmetric with respect to the origin. Therefore, the curve $(C_g)$ is symmetric with respect to the origin.

3. Final Answer

1) $D_g = [-3, 3]$

2) $g(-1) \approx -2$, $g(1) \approx 2$, $g(-1) = -g(1)$; $g(-3) = 0$, $g(3) = 0$, $g(-3) = g(3)$

3) $g(-x) = -g(x)$ for all $x \in D_g$.

4) The curve $(C_g)$ is symmetric with respect to the origin.