Worksheet on identifying even and odd functions with graphical and algebraic examples.

Let’s solve each part of the problem step by step.

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## 1. Indicate which of the following functions are even, odd, or neither.

We are given 4 graphs: (a), (b), (c), (d).

Recall:
- Even function: symmetric about the y-axis → f(-x) = f(x)
- Odd function: symmetric about the origin → f(-x) = -f(x)

Graph (a):

This is a straight line passing through the origin with negative slope (like y = -x).
→ Rotating 180° around origin maps the graph onto itself → symmetric about origin → Odd

✔ Answer: Odd

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Graph (b):

This is a downward-opening parabola symmetric about the y-axis (like y = -x² + c).
→ Reflecting over y-axis leaves graph unchanged → symmetric about y-axis → Even

✔ Answer: Even

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Graph (c):

This is a straight line NOT passing through the origin (has positive y-intercept and negative slope).
→ Not symmetric about y-axis (left/right not mirror images)
→ Not symmetric about origin (rotating 180° doesn’t map to itself)
→ So, Neither

✔ Answer: Neither

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Graph (d):

This is a V-shaped graph (absolute value-like) opening downward, vertex at origin? Wait — actually, it looks like two lines meeting at origin, symmetric about y-axis (like y = -|x|).
→ If you reflect over y-axis, left side mirrors right side → symmetric about y-axis → Even

✔ Answer: Even

> Note: Some might think this is odd because it passes through origin, but symmetry about origin requires that if (x,y) is on graph, so is (-x,-y). For example, point (1, -1) would require (-1, 1) to be on graph — but in this graph, (-1, -1) is there, not (1,1). So not odd.

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✔ Final Answers for #1:

- Graph (a): Odd
- Graph (b): Even
- Graph (c): Neither
- Graph (d): Even

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## 2. Algebraically determine whether each function is odd, even, or neither.

Use definitions:
- Even: f(-x) = f(x)
- Odd: f(-x) = -f(x)
- Otherwise: Neither

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a) f(x) = 3x³ - 5x² + 1

Compute f(-x):

f(-x) = 3(-x)³ - 5(-x)² + 1 = -3x³ - 5x² + 1

Compare to f(x) = 3x³ - 5x² + 1 → Not equal → Not even.

Compare to -f(x) = -3x³ + 5x² - 1 → Not equal → Not odd.

✔ Answer: Neither

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b) f(x) = |x|

f(-x) = |-x| = |x| = f(x) → Even

✔ Answer: Even

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c) f(x) = 12x⁴ + 6x³ - 2x

f(-x) = 12(-x)⁴ + 6(-x)³ - 2(-x) = 12x⁴ - 6x³ + 2x

Compare to f(x) = 12x⁴ + 6x³ - 2x → Not same → Not even.

Compare to -f(x) = -12x⁴ - 6x³ + 2x → Not same as f(-x) → Not odd.

✔ Answer: Neither

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d) f(x) = 4x³ - 7

f(-x) = 4(-x)³ - 7 = -4x³ - 7

Compare to f(x) = 4x³ - 7 → Not same → Not even.

Compare to -f(x) = -4x³ + 7 → Not same as f(-x) → Not odd.

✔ Answer: Neither

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e) f(x) = x² + 2x + 2

f(-x) = (-x)² + 2(-x) + 2 = x² - 2x + 2

Compare to f(x) = x² + 2x + 2 → Not same → Not even.

Compare to -f(x) = -x² - 2x - 2 → Not same → Not odd.

✔ Answer: Neither

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f) f(x) = (x² - 5)/(2x² + x)

First, note domain excludes x=0 and x=-1/2 (denominator zero), so we can test f(-x).

f(-x) = [(-x)² - 5] / [2(-x)² + (-x)] = (x² - 5)/(2x² - x)

Compare to f(x) = (x² - 5)/(2x² + x)

→ Not same → Not even.

Now check -f(x) = -(x² - 5)/(2x² + x) = (-x² + 5)/(2x² + x)

Compare to f(-x) = (x² - 5)/(2x² - x) → Not same → Not odd.

✔ Answer: Neither

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✔ Final Answers for #2:

a) Neither
b) Even
c) Neither
d) Neither
e) Neither
f) Neither

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## 3. The graphs of an odd function are symmetric about the origin. What geometric property characterizes even functions?

As defined at the top:

> Even functions are symmetric with respect to the y-axis.

So, geometrically:
→ If you fold the graph along the y-axis, the left and right sides match perfectly.

✔ Answer: Even functions are symmetric with respect to the y-axis.

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## 4. Suppose f is an odd function whose domain includes zero. Explain why f(0) = 0 must be true.

By definition of odd function:
f(-x) = -f(x) for all x in domain.

Since 0 is in domain, plug in x = 0:

f(-0) = -f(0)
→ f(0) = -f(0)
→ Add f(0) to both sides: 2f(0) = 0
→ f(0) = 0

✔ Answer: Because f(-0) = -f(0) ⇒ f(0) = -f(0) ⇒ 2f(0)=0 ⇒ f(0)=0.

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## 5. Can a function be both even and odd? (Hint: Consider constant functions of the form f(x) = c.)

Suppose f is both even and odd.

Then:
- f(-x) = f(x) [even]
- f(-x) = -f(x) [odd]

So, combining: f(x) = -f(x) ⇒ 2f(x) = 0 ⇒ f(x) = 0 for all x.

Thus, the only function that is both even and odd is the zero function: f(x) = 0.

Check:
- f(-x) = 0 = f(x) → even
- f(-x) = 0 = -0 = -f(x) → odd

✔ Answer: Yes, but only the zero function f(x) = 0. Any other constant function f(x) = c (c ≠ 0) is even but not odd, since f(-x) = c ≠ -c unless c=0.

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## ✔ Final Summary:

1. Graphs:

- (a) Odd
- (b) Even
- (c) Neither
- (d) Even

2. Algebraic:

- a) Neither
- b) Even
- c) Neither
- d) Neither
- e) Neither
- f) Neither

3. Even functions: symmetric about the y-axis

4. For odd f with 0 in domain: f(0) = 0 (because f(0) = -f(0))

5. Only possible if f(x) = 0 for all x — the zero function.

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