Graphs of Functions Classification $$ \qquad $$ Even Function Odd Function Neither Even nor Odd Function Both Even and Odd Function

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To classify functions based on their symmetry, we can use the following definitions:

1. **Even Function**: A function $$ f(x) $$ is even if for every $$ x $$ in the domain of $$ f $$, the following holds:
   $$
   f(-x) = f(x)
   $$
   This means the graph of the function is symmetric with respect to the y-axis.

2. **Odd Function**: A function $$ f(x) $$ is odd if for every $$ x $$ in the domain of $$ f $$, the following holds:
   $$
   f(-x) = -f(x)
   $$
   This means the graph of the function is symmetric with respect to the origin.

3. **Neither Even nor Odd Function**: If a function does not satisfy the conditions for being even or odd, it is classified as neither.

4. **Both Even and Odd Function**: The only function that is both even and odd is the zero function, $$ f(x) = 0 $$, since it satisfies both conditions:
   $$
   f(-x) = 0 = f(x) \quad \text{and} \quad f(-x) = 0 = -f(x)
   $$

### Steps to Classify a Function

1. **Identify the Function**: Start with the function you want to classify.

2. **Check for Evenness**: Substitute $$-x$$ into the function and see if it equals $$f(x)$$.

3. **Check for Oddness**: Substitute $$-x$$ into the function and see if it equals $$-f(x)$$.

4. **Determine the Classification**: Based on the results of the checks, classify the function as even, odd, neither, or both.

### Example

Let's classify the function $$ f(x) = x^2 $$:

1. **Check for Evenness**:
   $$
   f(-x) = (-x)^2 = x^2 = f(x) \quad \text{(Even)}
   $$

2. **Check for Oddness**:
   $$
   f(-x) = x^2 \quad \text{and} \quad -f(x) = -x^2 \quad \text{(Not Odd)}
   $$

3. **Classification**: Since $$ f(-x) = f(x) $$ and not equal to $$-f(x)$$, $$ f(x) = x^2 $$ is an **Even Function**.

If you have a specific function in mind, please provide it, and I can help classify it accordingly!
To classify a function as even, odd, neither, or both, follow these steps: 1. **Even Function**: If $$ f(-x) = f(x) $$ for all $$ x $$, it's even. 2. **Odd Function**: If $$ f(-x) = -f(x) $$ for all $$ x $$, it's odd. 3. **Neither**: If it doesn't satisfy both conditions. 4. **Both**: Only the zero function $$ f(x) = 0 $$ is both even and odd. For example, $$ f(x) = x^2 $$ is even because $$ f(-x) = x^2 = f(x) $$.

Graphs of Functions Classification \( \qquad \) Even Function Odd Function Neither Even nor Odd Function Both Even and Odd Function

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