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To determine the nature of a function, the first step involves substituting the variable \(x\) with \(-x\). This substitution is essential to test if a function is even or odd. After replacing \(x\) with \(-x\), you focus on the behavior of the resulting expression compared to the original function.
This means replacing every occurrence of \(x\) in the function with \(-x\).
For example, if we start with the function \(h(x) = 2x^2 + x^4\), then by substituting, it becomes \(h(-x) = 2(-x)^2 + (-x)^4\).
Simplifying, we find \(h(-x) = 2x^2 + x^4\).
Notice that a key role in this process plays understanding the properties of exponents:

• Even powers (e.g., \((-x)^2 = x^2\)) result in positive values, no matter the sign of \(x\).
• Odd powers (e.g., implies a change in the sign as \((-x)^3 = -x^3\)). However, in this function, \(x^4\) remains positive since 4 is even.

This substitution sets the foundation for further analysis to determine if a function is even or odd.

A function is considered even if it satisfies the condition \(f(-x) = f(x)\).
This means the function is symmetrical around the y-axis, showing that substituting \(x\) with \(-x\) results in the original function itself.
Using our function \(h(x) = 2x^2 + x^4\), once substituted, we found \(h(-x) = 2x^2 + x^4\), which is exactly the same as \(h(x)\).
Thus, \(h(-x) = h(x)\) confirming that the function is even.
Key Characteristics of Even Functions:

• Even functions exhibit symmetry about the y-axis. Graphically, this means if you fold the graph along the y-axis, it perfectly overlaps.
• The presence of only even powers in a polynomial function, as in this exercise, often indicates an even function.
• Common examples include functions such as \(cos(x)\) and \(x^2\).

Recognizing these features in a function can help easily identify its evenness.

To tell if a function is odd, the condition \(f(-x) = -f(x)\) must hold true. This means swapping \(x\) with \(-x\) should yield the negative of the original function.
If this is true, the function will be symmetrical about the origin.
Let's check it out with \(h(x) = 2x^2 + x^4\).
After we substitute and simplify to find \(h(-x) = 2x^2 + x^4\), we compare it to \(-h(x) = -(2x^2 + x^4)\).
Notice here, \(h(-x)\) is not equal to \(-h(x)\), meaning \(h(x)\) is not an odd function.
Key Characteristics of Odd Functions:

• Odd functions show symmetry about the origin. Graphically, if you rotate the graph 180 degrees around the origin, it overlaps perfectly.
• This type of behavior is often observed when a function includes terms with only odd powers, like \(x^3\) or \(x^1\).
• Classic examples include \(sin(x)\) and \(x^3\).

Knowing these traits can help identify odd functions efficiently.