📖 Pre-Calculus – Even and Odd Functions | Grade 11

Pre-Calculus

Even and Odd Functions

🔍 Why Does This Matter?

When a function is even or odd, it has special symmetry.

This allows us to:

Draw only half the graph and complete the rest
Simplify calculations (especially integrals)
Better understand the behaviour of the function

🪞 Even Function

\(f(-x) = f(x)\)

The value of the function at x equals its value at (−x)

🔑 Geometric meaning:

The graph is symmetric about the y-axis

If we fold the graph along the y-axis, both halves match exactly

📚 Examples of even functions:

Function	Verification
\(f(x) = x^2\)	\(f(-x) = (-x)^2 = x^2 = f(x)\) ✓
\(f(x) = x^4\)	\(f(-x) = (-x)^4 = x^4 = f(x)\) ✓
\(f(x) = \|x\|\)	\(f(-x) = \|-x\| = \|x\| = f(x)\) ✓
\(f(x) = \cos(x)\)	\(f(-x) = \cos(-x) = \cos(x) = f(x)\) ✓
\(f(x) = x^2 + 3\)	\(f(-x) = (-x)^2 + 3 = x^2 + 3 = f(x)\) ✓

💡 Rule of thumb: even powers of x (like x², x⁴, x⁶…) are even functions!

🔄 Odd Function

\(f(-x) = -f(x)\)

The value of the function at (−x) is the negative of its value at x

🔑 Geometric meaning:

The graph is symmetric about the origin (0,0)

If we rotate the graph 180° about the origin, it remains unchanged

📚 Examples of odd functions:

Function	Verification
\(f(x) = x\)	\(f(-x) = -x = -f(x)\) ✓
\(f(x) = x^3\)	\(f(-x) = (-x)^3 = -x^3 = -f(x)\) ✓
\(f(x) = x^5\)	\(f(-x) = (-x)^5 = -x^5 = -f(x)\) ✓
\(f(x) = \sin(x)\)	\(f(-x) = \sin(-x) = -\sin(x) = -f(x)\) ✓
\(f(x) = \frac{1}{x}\)	\(f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x)\) ✓

💡 Rule of thumb: odd powers of x (like x, x³, x⁵…) are odd functions!

⚠️ Special property: if an odd function is defined at x=0, then:

\(f(0) = 0\)

Because: \(f(-0) = -f(0)\) → \(f(0) = -f(0)\) → \(2f(0) = 0\)

🔬 How to Test Even/Odd?

Step 1: Compute \(f(-x)\)

Substitute (−x) for x everywhere in the function

Step 2: Compare the result

If \(f(-x) = f(x)\) → even
If \(f(-x) = -f(x)\) → odd
If neither → neither even nor odd

✏️ Detailed Examples

Example 1: Check whether \(f(x) = x^4 - 3x^2 + 1\) is even/odd

Solution:

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

\(= x^4 - 3x^2 + 1\)

\(= f(x)\)

✓ The function is even

Example 2: Check whether \(f(x) = x^3 - 2x\) is even/odd

Solution:

\(f(-x) = (-x)^3 - 2(-x)\)

\(= -x^3 + 2x\)

\(= -(x^3 - 2x)\)

\(= -f(x)\)

✓ The function is odd

Example 3: Check whether \(f(x) = x^2 + x\) is even/odd

Solution:

\(f(-x) = (-x)^2 + (-x) = x^2 - x\)

Let us check:

\(f(x) = x^2 + x\) → \(f(-x) \neq f(x)\)

\(-f(x) = -x^2 - x\) → \(f(-x) \neq -f(x)\)

✗ The function is neither even nor odd

⚖️ Comparison Table

	Even function	Odd function
Definition	\(f(-x) = f(x)\)	\(f(-x) = -f(x)\)
Symmetry	About the y-axis	About the origin (0,0)
Examples	\(x^2, x^4, \|x\|, \cos x\)	\(x, x^3, x^5, \sin x\)
Powers	Even powers (2, 4, 6…)	Odd powers (1, 3, 5…)
f(0)	Can be any value	Must be 0 (if defined)

🧮 Additional Properties

Sum of even functions: even + even = even
Sum of odd functions: odd + odd = odd
Product of even functions: even × even = even
Product of odd functions: odd × odd = even
Mixed product: even × odd = odd

💡 The only function that is both even and odd:

\(f(x) = 0\)

📝 Summary

Even: \(f(-x) = f(x)\) → symmetry about the y-axis

Odd: \(f(-x) = -f(x)\) → symmetry about the origin

Even power → even function | Odd power → odd function

Worked Examples

Given \(x^2\). Is the function even, odd, or neither?

Show solution

The function is even.

✓ Correct

Odd

Neither

Cannot be determined

Solution: \(f(-x)=(-x)^2=x^2=f(x)\) → even.

Given \(x^3\). Is the function even, odd, or neither?

Show solution

The function is odd.

✓ Correct

Even

Neither

Cannot be determined

Solution: \(f(-x)=(-x)^3=-x^3=-f(x)\) → odd.

Given \(x^2+3\). Is the function even, odd, or neither?

Show solution

The function is even.

✓ Correct

Odd

Neither

Cannot be determined

Solution: \(f(-x)=(-x)^2+3=x^2+3=f(x)\) → even.

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