If f(x) is an odd function, then the curve y=f(x) is symmetric

Options

A. about x-axis
B. about y-axis
C. about both the axes
D. in opposite quadrants

Answer

To determine the symmetry of the curve \( y = f(x) \) when \( f(x) \) is an odd function, we will follow these steps:

### Step 1: Understand the Definition of an Odd Function
An odd function is defined by the property:
\[ f(-x) = -f(x) \]
for all \( x \) in the domain of \( f \). This means that the value of the function at \( -x \) is the negative of the value of the function at \( x \).

**Hint:** Recall that for an odd function, the output changes sign when the input is negated.

### Step 2: Analyze the Implications of the Definition
From the definition, if we take a point \( (a, f(a)) \) on the curve, then the corresponding point for \( -a \) will be:
\[ (-a, f(-a)) = (-a, -f(a)) \]
This means that if \( (a, f(a)) \) is in the graph, then \( (-a, -f(a)) \) is also in the graph.

**Hint:** Consider how the coordinates change when you reflect them across the origin.

### Step 3: Visualize the Symmetry
The points \( (a, f(a)) \) and \( (-a, -f(a)) \) indicate that the graph of the function is symmetric with respect to the origin. This means that if you were to rotate the graph 180 degrees around the origin, it would look the same.

**Hint:** Think about how rotating a point in the Cartesian plane affects its coordinates.

### Step 4: Conclusion
Since the graph of \( y = f(x) \) contains both \( (a, f(a)) \) and \( (-a, -f(a)) \) for every \( a \), we conclude that the curve is symmetric about the origin.

Thus, the correct answer is that the curve \( y = f(x) \) is symmetric in the opposite quadrant.

**Final Answer:** The curve \( y = f(x) \) is symmetric about the origin (option D).