Answer
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 \( f(-x) = x^2 = f(x) \).
Solution
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!
Answered by UpStudy AI and reviewed by a Professional Tutor
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