Which of the following functions is an odd function?

Options

A.
B.
C.
D.

Answer

To determine which of the given functions 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:
\[ f(-x) = -f(x) \]
This means that for every point \( x \) in the domain of the function, the value of the function at \( -x \) is the negative of the value at \( x \).

### Step 2: Analyze the Graphs
We have four options (graphs) to analyze. We need to check each graph for symmetry about the origin. A graph is symmetric about the origin if, when you rotate it 180 degrees around the origin, it looks the same.

### Step 3: Check Each Option
- **Option 1**: Check if the graph is symmetric about the origin. If it is not, it cannot be an odd function.
- **Option 2**: Check if the graph is symmetric about the y-axis. If it is, it is an even function, not an odd function.
- **Option 3**: Check for symmetry. If there is no symmetry about the origin or y-axis, it is neither odd nor even.
- **Option 4**: Check if the graph is symmetric about the origin. If it is, then this graph represents an odd function.

### Step 4: Identify the Odd Function
After analyzing the graphs:
- **Option 4** is the only graph that is symmetric about the origin, satisfying the condition for an odd function.

### Conclusion
Thus, the function represented by **Option 4** is an odd function.

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