Practice problems for identifying even and odd functions through graphical and algebraic methods.

Let's go through the Even and Odd Functions Practice Problems step by step, explaining how to determine whether a function is even, odd, or neither, both graphically (Section A) and algebraically (Section B).

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🔷 Definitions:

- A function $ f(x) $ is even if:
$$
f(-x) = f(x)
$$
→ The graph is symmetric about the y-axis.

- A function $ f(x) $ is odd if:
$$
f(-x) = -f(x)
$$
→ The graph is symmetric about the origin (180° rotation).

- If neither condition holds, the function is neither even nor odd.

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## ✔ Section A: Graphical Analysis

We analyze each graph based on symmetry.

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1. First Graph

- The graph appears to be symmetric about the origin.
- For example, if you rotate it 180° around the origin, it maps onto itself.
- Points like $ (a, b) $ have corresponding points $ (-a, -b) $.
- This is characteristic of an odd function.

✔ Answer: Odd

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2. Second Graph

- This graph is a parabola-like shape opening upward, but shifted right.
- It’s not symmetric about the y-axis (because the vertex is not at $ x=0 $), so it's not even.
- It’s also not symmetric about the origin — for example, $ f(1) \neq -f(-1) $, and values don’t reflect through the origin.
- So it fails both conditions.

✔ Answer: Neither

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3. Third Graph

- This is a parabola opening downward with its vertex at the origin.
- Symmetric about the y-axis: $ f(-x) = f(x) $
- Example: $ f(2) = f(-2) $, etc.

✔ Answer: Even

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## ✔ Section B: Algebraic Analysis

We will use the definitions:
- Compute $ f(-x) $
- Compare with $ f(x) $ and $ -f(x) $

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1. $ f(x) = x^3 - x^2 + 4x + 2 $

Compute $ f(-x) $:
$$
f(-x) = (-x)^3 - (-x)^2 + 4(-x) + 2 = -x^3 - x^2 - 4x + 2
$$

Compare:
- Is $ f(-x) = f(x) $? No.
- Is $ f(-x) = -f(x) $? Let's check $ -f(x) $:
$$
-f(x) = -(x^3 - x^2 + 4x + 2) = -x^3 + x^2 - 4x - 2
$$
Not equal to $ f(-x) $

✘ Neither

✔ Answer: Neither

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2. $ f(x) = -x^2 + 10 $

$$
f(-x) = -(-x)^2 + 10 = -x^2 + 10 = f(x)
$$

→ $ f(-x) = f(x) $ → Even

✔ Answer: Even

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3. $ f(x) = x^3 + 4x $

$$
f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x = -(x^3 + 4x) = -f(x)
$$

→ $ f(-x) = -f(x) $ → Odd

✔ Answer: Odd

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4. $ f(x) = -x^3 + 5x - 2 $

$$
f(-x) = -(-x)^3 + 5(-x) - 2 = -(-x^3) - 5x - 2 = x^3 - 5x - 2
$$

Now compare:
- $ f(x) = -x^3 + 5x - 2 $
- $ f(-x) = x^3 - 5x - 2 $
- $ -f(x) = x^3 - 5x + 2 $

Is $ f(-x) = f(x) $? No
Is $ f(-x) = -f(x) $? No (because constant term is $-2$ vs $+2$)

✘ Neither

✔ Answer: Neither

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5. $ f(x) = \sqrt{x^4 - x^2 + 4} $

Note: $ x^4 $ and $ x^2 $ are even powers.

$$
f(-x) = \sqrt{(-x)^4 - (-x)^2 + 4} = \sqrt{x^4 - x^2 + 4} = f(x)
$$

→ $ f(-x) = f(x) $ → Even

✔ Answer: Even

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6. $ f(x) = |x + 4| $

$$
f(-x) = |-x + 4| = |-(x - 4)| = |x - 4|
$$

Compare:
- $ f(x) = |x + 4| $
- $ f(-x) = |x - 4| $
- These are not equal → not even
- $ -f(x) = -|x + 4| $, which is negative, while $ f(-x) $ is non-negative → not odd

✘ Neither

✔ Answer: Neither

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7. $ f(x) = |x| + 4 $

$$
f(-x) = |-x| + 4 = |x| + 4 = f(x)
$$

→ $ f(-x) = f(x) $ → Even

✔ Answer: Even

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8. $ f(x) = x^4 - 2x^2 + 4 $

All terms have even powers.

$$
f(-x) = (-x)^4 - 2(-x)^2 + 4 = x^4 - 2x^2 + 4 = f(x)
$$

→ Even

✔ Answer: Even

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9. $ f(x) = \sqrt[3]{x} $

Cube root is defined for all real numbers.

$$
f(-x) = \sqrt[3]{-x} = -\sqrt[3]{x} = -f(x)
$$

→ $ f(-x) = -f(x) $ → Odd

✔ Answer: Odd

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10. $ f(x) = x\sqrt{x^2 - 1} $

Check domain first: $ x^2 - 1 \geq 0 $ → $ |x| \geq 1 $

Now compute $ f(-x) $:
$$
f(-x) = (-x)\sqrt{(-x)^2 - 1} = (-x)\sqrt{x^2 - 1} = -x\sqrt{x^2 - 1} = -f(x)
$$

→ $ f(-x) = -f(x) $ → Odd

✔ Answer: Odd

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## ✔ Final Answers Summary

Section A (Graphs):

1. Odd
2. Neither
3. Even

Section B (Algebra):

1. Neither
2. Even
3. Odd
4. Neither
5. Even
6. Neither
7. Even
8. Even
9. Odd
10. Odd

These match the provided answers!

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🧠 Key Takeaways:

- Even functions: Symmetric about y-axis → $ f(-x) = f(x) $
- Odd functions: Symmetric about origin → $ f(-x) = -f(x) $
- Polynomials: Only even-powered terms → even; only odd-powered terms → odd; mixed → neither
- Absolute value & roots: Check behavior under $ -x $
- Always test $ f(-x) $ algebraically when possible

Let me know if you'd like visual explanations or graphs!