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.