functions: even/odd/neither

Math I: Unit 5 (Part 2)

A function is even…

If the graph is symmetrical about the y-axis,then it’s even. **Fold hotdog!

A function is odd…

If the graph is symmetrical about the y-axis &x-axis (or symmetrical about the origin),then it’s odd. **Fold hotdog & hamburger!

A function is even if f(-x) = f(x)

Example 1: f(x) = 2x2 + 5

If you substitute in -x and get the SAME function that you started with, then it’s even.

f(-x)= f(x)Replace x with –x 2(-x)2

+ 5 2x2 + 5

Simplify 2(-x)(-x) + 5

2x2 + 5

The new equation 2x2 + 5

2x2 + 5The equations are exactly the

SAME…so EVEN function.

A function is odd if f(-x) = -f(x)

If you substitute in -x and get the OPPOSITE function(all the signs change),then it’s odd.

Example: f(x) = 4x3 + 2x

f(-x) f(x)Replace x with –x 4(-x)3 +

2(-x) 4x3 + 2x

Simplify 4(-x)(-x)(-x) + 2(-x)

4x3 + 2x

The new equation -4x3 – 2x

4x3 + 2xEVERY sign changed…so OPPOSITES…

ODD function

Graphically…If a function does not have y-axis symmetry OR origin symmetry…then it has NEITHER.

Algebraically…If, after substituting –x in place of x, the equation is not EXACTLY the same OR complete OPPOSITES, then the function is NEITHER.

Neither Even Odd

f(x) = x4 + x2

f(x) = 1 + x3

f(x) = 2x3 + x

f(-x) f(x)

(-x)4 + (-x)2 x4 + x2

(-x)(-x)(-x)(-x) + (-x)(-x)

x4 + x2

x4 + x2

x4 + x2

f(-x) f(x)

1 + (-x)3 1 + x3

1 + (-x)(-x)(-x)

1 + x3

1 – x3 1 + x3

f(-x) f(x)

2(-x)3 + (-x)

2x3 + x

2(-x)(-x)(-x) + (-x)

2x3+ x

-2x3 - x

2x3+ x

SAME – so EVEN

OPPOSITES– so ODD

Not same and Not all signs changed – so

NEITHER