Pc Functions Odd Even

Symmetric about the y axis

FUNCTIONS Symmetric about the origin

Even functions have y-axis Symmetry 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7

So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

Odd functions have origin Symmetry 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7

So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

x-axis Symmetry We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7

A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even.

f ( x) = 5x − 2 x + 1 4

2

Is this function even?

YES

f ( − x ) = 5(− x) − 2(− x) + 1 = 5 x − 2 x + 1 4

2

4

2

f ( x ) = 2 x − x Is this function even? NO 3 3 f ( − x ) = 2(−x) − (−x) = −2 x + x 3

A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd.

f ( x) = 5x − 2 x + 1 4

2

Is this function odd?

NO

f ( − x ) = 5(− x) − 2(− x) + 1 = 5 x − 2 x + 1 4

2

4

2

f ( x ) = 2 x − x Is this function odd? YES 3 3 f ( − x ) = 2(−x) − (−x) = −2 x + x 3

If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all 3 terms didn’t change signs, so NEITHER.

f ( x) = 5x − 1

f ( − x ) = 5( − x ) − 1 = − 5 x − 1 3

3

f ( x ) = −3 x − x + 2 4

2

Got f(x) back so EVEN.

f ( − x ) = −3(− x) − (− x) + 2 = −3 x − x + 2 4

2

4

2