The function f(x) = |x| + x ^ 2 + x ^ 4 will be?even or odd  ?

To determine whether the function \( f(x) = |x| + x^2 + x^4 \) is even, odd, or neither, we need to analyze its symmetry properties.

### Definitions of Even and Odd Functions

- A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \) for all \( x \).

### Checking for Evenness

Let’s compute \( f(-x) \) and compare it with \( f(x) \):

\[
f(x) = |x| + x^2 + x^4
\]

Now, compute \( f(-x) \):

\[
f(-x) = |-x| + (-x)^2 + (-x)^4
\]

Simplifying each term:

1. **Absolute Value Term**: \( |-x| = |x| \)
2. **Square Term**: \( (-x)^2 = x^2 \)
3. **Quartic Term**: \( (-x)^4 = x^4 \)

So, 

\[
f(-x) = |x| + x^2 + x^4
\]

This simplifies to:

\[
f(-x) = f(x)
\]

### Conclusion

Since \( f(-x) = f(x) \) for all \( x \), the function \( f(x) = |x| + x^2 + x^4 \) is an **even function**.

Would you like further explanation or have any more questions about the function?

### Additional Questions:

1. Why is it important to check each term separately when determining if a function is even or odd?
2. Can we find the symmetry of other common functions, such as \( x^3 \) or \( \sin(x) \)?
3. What would make a function neither even nor odd?
4. How would the symmetry change if we modified the function, say to \( |x| + x^3 + x^4 \)?
5. How can the graph of an even function help visually confirm its symmetry?

**Tip:** For any function involving powers of \( x \), even functions will have only even powers (or absolute values), while odd functions will have only odd powers.

Explore whether the function f(x) = |x| + x^2 + x^4 is even, odd, or neither with a detailed analysis of its symmetry properties and mathematical reasoning. Suitable for high school students and anyone interested in function symmetry and polynomial functions.

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