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MHF4U_2011: Advanced Functions, Grade 12, University Preparation

Unit 2: Advanced Polynomial and Rational Functions

Activity 3: Characteristics of Odd and Even functions

Homework

For each of the following functions, classify each as: even, odd or neither. You must show your work to prove your classification. If you are experiencing difficulty, contact your teacher.

1. Show a proof algebraically

• f ( x) = x ² − 2 x

• f ( x) = 3x ⁵ − 4 x

2. Decide if these are even or odd or neither from the graph

b)

f)

c)

g)

d)

h)

Evens and Odds – Practice

Determine whether each of the functions below is even, odd or neither. Justify your answers.

3.

4.

5.

f(x) = 3x² + 4

6.

f(x) = -2x + 5

7.

f(x) = 2x² + 3x

8.

f(x) = -3x³ + x

Homework Solutions

For each of the following functions, classify each as: even, odd or neither. You must show your work to prove your classification. If you are experiencing difficulty, contact your teacher.

1.

a)

• ( x ) = x ² − 2 x

•  ( − x ) = ( − x ) ² − 2( − x ) = x ² + 2 x − f ( x ) = − ( x ² − 2 x ) = − x ² + 2 x
• ( x ) ≠ f ( − x ) ≠ − f ( x )

∴ f ( x) is neither.

c)

• ( x ) = 3 x ⁵ − 4 x

•  ( − x ) = 3( − x ) ⁵ − 4( − x ) = −3 x ⁵ + 4 x − f ( x ) = − (3 x ⁵ − 4 x ) = − 3 x ⁵ + 4 x
• ( − x ) = − f ( x)

∴ f ( x ) is odd.

c)

• ( x ) = f (− x)

∴ f ( x) is even.

d)

x

f ( x) =

x² − 3

• ( x ) ≠ f (− x ) ≠ − f ( x)

∴ f ( x) is neither.

2. Decide if these are even or odd or neither from the graph

b)

Even

f)

Neither

c)

Neither

g)

Neither

d)

Even

h)

odd

Evens and Odds – Practice

Determine whether each of the functions below is even, odd or neither. Justify your answers.

3.Even

4.Odd

5.

7.