Classify the following functions for being even or odd:
`x^2 - |x|`

Answer

To classify the function \( f(x) = x^2 - |x| \) as even or odd, we will follow these steps:

### Step 1: Understand the definitions
A function \( f(x) \) is classified as:
- **Even** if \( f(-x) = f(x) \) for all \( x \).
- **Odd** if \( f(-x) = -f(x) \) for all \( x \).

### Step 2: Calculate \( f(-x) \)
We need to find \( f(-x) \):

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

### Step 3: Simplify \( f(-x) \)
Now, simplify the expression:

1. Calculate \( (-x)^2 \):
\[ (-x)^2 = x^2 \]

2. Calculate \( |-x| \):
\[ |-x| = |x| \]

Putting it all together:
\[ f(-x) = x^2 - |x| \]

### Step 4: Compare \( f(-x) \) with \( f(x) \)
Now we compare \( f(-x) \) with \( f(x) \):

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

Since \( f(-x) = f(x) \), we conclude that the function is even.

### Step 5: Conclusion
Thus, the function \( f(x) = x^2 - |x| \) is classified as an **even function**.

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