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Even Odd Functions

Even Functions

Algebraically, a function is considered an even function if f(x) = f(-x) for all x's in the equation. This means that if you substitute –x for x in the original equation and get what you started with, the function is an even function.

If a function contains only even-numbered exponents, the function is considered an even function.

For example, suppose you had f(x) = x2. This doesn't have a constant but it does have an even-numbered exponent. Therefore, this is an even function.

In fact, all standard functions whose exponent is even is considered an even function. They are x4, x6, x8and so on.

Odd Function

An odd function is a function where –f(x) = f(-x). This means that if you substitute -x into the function and you get the opposite of the function, it is odd. If a function contains only odd-numbered exponents and no constants, the function is an odd function.

Suppose you have the function f(x) = x3. This does not have a constant and the exponent is odd, so this is an example of an odd function. This also holds true for f(x) = x5, x7, x9, etc.

Neither Even or Odd

What if a function is neither even nor odd?

A function will be neither even nor odd if it doesn't meet the conditions you previously learned. A function doesn't necessarily have to be even or odd because these are just names of functions.

In fact, most functions will be neither even nor odd.

This information will be important because it will give you a way to algebraically check to see if a function is even, odd, or neither. Remember:

  • if f(x) = f(-x), then the function is even.

  • if -f(x) = f(-x), then the function is odd.

Another Way of Thinking About Even Function

Another way of thinking about an even function:

Start with the statement "If f(x) = f(-x), then the function is even." This means that if you substitute -x, simplify, and you find that it is equal to the same function you began with, then the function is considered even.

Another Way of Thinking About Odd Function

Another way of thinking about an odd function:

The other statement says that "If -f(x) = f(-x), then the function is odd." This means that if you substitute -x, simplify, and you find that it is the opposite of the original function, then the function is considered odd.

Algebraic Examples

Example #1

Watch Determine Whether a Given Function is Even or Odd Given Its Equation.

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Example #2

Watch Determine Whether a Given Function is Even or Odd Given Its Equation.

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Example #3

Watch Determine Whether a Given Function is Even or Odd Given Its Equation.

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Graphic Examples

Even/Odd Graphically

You will not always be given the equation for the function. Often times, you might need to identify if a function is even or odd from its graph.

If a function is even, its graph will be symmetrical across the y-axis.

If a function is odd, its graph will be symmetrical across the origin.

Symmetrical Across the Y-Axis

You can easily identify if a graph is symmetrical across the y-axis. For example, the graphs below are all symmetrical across the y-axis.

The graph of a parabola with a vertex at (0,0) opening up with points (negative 1, 1) and (1,1). If you flip the graph over the y-axis, it will look exactly the same. The graph of a parabola with a vertex at (0, negative 3) opening up. If you flip the graph over the y-axis, it will look exactly the same. The graph of a parabola with a vertex at (0,4 ) opening down. If you flip the graph over the y-axis, it will look exactly the same.

Example #4

Watch Determine Whether a Function is Symmetric across the Y-Axis Given Its Graph.

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Symmetrical About the Origin

Each graph below is symmetrical about the origin. An easy way to tell that a graph is symmetrical about the origin is if you reflect it about one axis and then the other, it will land on top of itself.

A graph of a polynomial that starts down and ends up. The graph comes from negative infinity up through the x-axis between negative 2 and negative 1, goes up to (negative 1, 1) then turns back down and goes through (0,0), continues to go down to approximately (3 fourths, negative 1), turns back up and goes through the x-axis between 1 and 2, then continues going up to positive infinity. The 1st and 4th quadrants look the same but with opposite signs for each point and the 2nd and 3rd quadrants look the same but with opposite signs for each point. A graph of y = 1 over x. There is a graph increasing in the second quadrant and a graph increasing in the fourth quadrant. A graph of a polynomial that starts up and ends down. The graph comes from positive infinity down through the x-axis between negative 2 and negative 1, goes down to approximately (negative 3 fourths, negative 3 halves) then turns back up and goes through (0,0), continues to go up to approximately (3 fourths, 3 halves), turns back down and goes through the x-axis between 1 and 2, then continues going down to negative infinity. The 1st and 4th quadrants look the same but with opposite signs for each point and the 2nd and 3rd quadrants look the same but with opposite signs for each point.

Example #5

Watch Determine Whether a Function is Symmetric about the Origin Given Its Graph.

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Neither Even or Odd

Now that you know how to tell if a function is even or odd based on the graph of the function, what if a graph is not a function like the ones below?

The graph of the positive square root function going up and to the right and the negative square root function going down and to the right, both with a vertex at (0,0) on the same coordinate plane . The graph looks like a sideways U going to the right. A graph of a circle with a center at (0,0) and radius 2. A graph of a sideways u with a vertex at (2,0) opening to the right and another sideways U opening to the left with a vertex at (negative 2, 0).

If the graph is not a function, then it is considered neither even nor odd.

Example #6

Watch Determine Whether a Function is Even, Odd, or Neither Given Its Graph.

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Example #7

Watch Determine Whether a Function is Even, Odd, or Neither Given Its Graph.

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