Symmetry of Functions

Even, Odd, or Neither?

Even Functions

• All exponents are even.

• May contain a constant.

• f(x) = f(-x)

• Symmetric about the y-axis

All even exponents

• Example:

4 2( ) 5 3f x x x

Both exponents are even. It does not matter what the coefficients are.

May Contain a Constant

• Example

6 2( ) 3 7 1f x x x

Even exponents (coefficients don’t matter)

Constant does not affect even function.

f(x) = f(-x)

• Given f(x) = 5x² - 7, find f(-x) to determine if f(x) is even, odd, or neither.

1) Substitute –x for x.

2) f(-x) = 5(-x)² - 7 = 5x² -7

3) Because f(x) = f(-x),

f(x) = 5x² - 7 is an even function.

f(x) = f(-x)

• Given f(x) = 4x² - 2x + 1, determine if f(x) is even, odd, or neither.

1) Substitute –x for x.

2) f(-x) = 4(-x)² - 2(-x) + 1 = 4x² +2x + 1

3) f(-x) ≠ f(x)

4) Therefore, f(x) is NOT an even function. (We will revisit to determine what it is.)

Symmetric About the y-axis

• The following are symmetric about the y-axis.

Odd Functions

• Only odd exponents.

• NO constants!

• f(-x) = -f(x)

• Symmetric about the origin.

All Odd Exponents

• Example

5 3( ) 8 6 4f x x x x

All odd exponents.

Understood 1 exponent

NO Constants• Example:

5 3( ) 6 8 5f x x x Odd exponents

NO constants in odd functions!

f(-x) = -f(x)

• Given f(x) = 4x³ + 2x, find f(-x) and f(-x) to determine if f(x) is even, odd, or neither.

• f(-x) = 4(-x)³ + 2(-x) = -4x³ - 2x

• -f(x) = -4x³ - 2x

• Because f(-x) = -f(x),

f(x) is an odd function.

f(-x) = -f(x)• Given f(x) = 5x³ + 7x², find f(-x) and

f(-x) to determine if f(x) is even, odd, or neither.

• f(-x) = 5(-x)³ + 7(-x)² = -5x³ + 7x²• -f(x) = -5x³ - 7x²• f(-x) ≠ -f(x), therefore f(x) is NOT an odd

function.

Symmetric About the Origin

• These graphs are symmetric about the origin.

Neither?

• Mixture of even and odd exponents.

• All odd exponents with a constant.

• f(x) ≠ f(-x) AND f(-x) ≠ -f(x)

Examples of Neither• f(x) = 4x³ - 5x²

• f(x) = 5x³ + 7

Mixture of odd and even exponents.

Odd exponents with a constant.

Examples of Neither• If f(x) = -3x³ + 2x², determine if f(x) is

even, odd, or neither.

1) Find f(-x).

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

3) Find –f(x).

4) -f(x) = 3x³ - 2x²

5) Because f(-x) ≠ f(x) and f(-x) ≠ -f(x), f(x) is neither even nor odd.