Even and Odd Functions
When analyzing functions, you will notice some of them may be symmetrical. We use a special type of function notation to label some of these functions, naming them even, odd, or neither. A function can be labeled even if f(x) = f(-x), meaning if you plug –x into the function and simplify, you will get the exact same function. For odd functions, f(x) = -f(x), meaning if you plug –x into the function and simplify, you will get the exact opposite of the original function. A function will be labeled neither if it is neither even nor odd. Remember we are only analyzing true functions.
Even Functions
Remember a function is even if f(x) = f(-x). Let’s examine this function: f(x) = 4x4 + 6x2
To determine if the function is even, odd, or neither, I will first test to see if it is even by plugging in –x into the function and simplifying it.
f(x) = 4x4 + 6x2
f(-x) = 4(-x)4 + 6(-x)2 Simplify. Remember (-x)4 = (-x)(-x)(-x)(-x) = x4
f(-x) = 4x4 + 6x2
Therefore, f(x) = f(-x)
So as you can see we get the original function after plugging in –x and simplifying. Let’s see what the graph of f(x) = 4x4 + 6x2 looks like.
Since even and odd functions are symmetrical, we notice there is symmetry about the y-axis. It is important to note that even functions will show symmetry about the y-axis. Let’s look at the graph of another even function, f(x) = -2x2 + 8.
Without doing any work, we notice the graph shows symmetry about the y-axis, therefore making it an even function. We can check this by seeing if f(x) = f(-x).
f(x) = -2x2 + 8
f(-x) = -2(-x)2 + 8 Simplify. Remember (-x)2 = (-x)(-x) = x2
f(-x) = -2x2 + 8
Therefore, f(x) = f(-x), which proves the function is even.
Remember, even functions will show symmetry about the y-axis only and f(x) = f(-x). If the graph shows symmetry about the x-axis, it is not a function because it will not pass the vertical line test.
Odd Functions
Remember a function is odd if f(x) = -f(x). Let’s examine the function: f(x) = 2x3 + x
To determine if the function is even, odd, or neither, I will first test to see if it is even by plugging –x into the function and simplifying it.
f(x) = 2x3 + x
f(-x) = 2(-x)3 + x Simplify. Remember (-x)3 = (-x)(-x)(-x) = (-x)3
f(-x) = -2x3 + (-x) or f(-x) = -2x3 – x
Therefore, f(x) f(-x). In fact, it is the exact opposite as 2x3 because -2x3 and x became –x.
Since it is the exact opposite, we can say f(x) = -f(x). If every term in the function becomes its opposite, then we can say the function is odd. This means the “positive terms” become “negative terms” and the “negative terms” become “positive terms.” Let’s see what the graph of f(x) = -2x2 + 8 looks like.
Notice the graph show symmetry about the origin, therefore making it an odd function. Let’s look of the graph of another odd function, f(x) = -2x5 + 2x3.
Without doing any work, we notice the graph shows symmetry about the origin. We can check this by seeing if f(x) = -f(x).
f(x) = -2x5 + 2x3
f(-x) = -2(-x)5 + 2(-x)3 Simplify.
f(-x) = 2x5 + (-2x)3 or f(-x) = 2x5 – 2x3
Therefore, if f(x) = -f(x) because each the function became its exact opposite. Each term became its opposite.
It is important to note that all terms must become their opposite for the function to be odd.
Neither Functions
Remember functions that are neither even nor odd will be classified as neither. Let’s examine the function: f(x) = -2x2 + 8x.
Notice the graph does not show symmetry about the y-axis or the origin, therefore it is neither even nor odd. We can check this by plugging –x into the function.
f(x) = -2x2 + 8x
f(-x) = -2(-x)2 + 8(-x) Simplify
f(-x) = -2x2 + (-8x) or f(-x) = -2x2 – 8x
Notice f(x) ≠ f(-x) because 8x became -8x and f(x) ≠ -f(x) because all terms did not become their opposite (-2x2 stayed the same and 8x became its opposite). Therefore, the function is neither even nor odd.
Domain and Range
Since even and odd functions classify as functions, we can use the tips for finding domain and range.
Domain is all x numbers that allow the function to work
Range is all y numbers that result from the input of an x number into the function.
The domain of functions can be any real number. This can be written as -∞ < x < ∞
The range of odd degree functions (determined by exponent associated with first term) can be any real number. This can be written as -∞ < ⅟ < ∞
The range of even degree functions (determined by exponent associated with first term) can vary. They can be written as -∞ < ⅟ ≤ global maximum or global minimum ≤ ⅟ < ∞
Roots or Zeros
The roots of polynomial equations can tell us the x-intercepts of the equation. We know that an x- intercept is when the graph crosses the x-axis, therefore the y value must equal zero. It is important to note the roots of a polynomial equation can consist of real or imaginary numbers. Real number roots will appear on the graph; however imaginary number roots will not be seen on the graph.
An easy way to determine the number of roots a polynomial has is to look at the degree (exponent) of the polynomial. The degree (exponent) associated with the first term of a polynomial will tell us the number of roots a polynomial has. For example:
This is the graph of the polynomial f(x) = 2x2 + 4x. We see that the degree (exponent) of the polynomial function is 2 because the first term has an exponent of 2. This means the polynomial should have two roots. As we can see by the graph, the polynomial crosses or touches the x-axis twice, once at 0 and once at -2.
We can test our roots to see if they are correct by plugging them into the polynomial and solving. If these are the correct roots, we should get answers of zero each time.
f(0) = 2(0)2 + 4(0) f(-2) = 2(-2)2 + 4(-2)
f(0) = 0 + 0 f(-2) = 8 + -8
f(0) = 0 f(-2) = 0
As we can see, each function gives us an answer of zero. Therefore, zero and -2 must be our roots.
There is another method for calculating the roots without looking at the graph. We will use the same function, f(x) = 2x2 + 4x. We can factor the polynomial, and since it is quadratic we can use the quadratic formula to factor.
x=-b±b2-4ac2a
x=-4±4-42(0)2(2)
x=-4±164
x=-4±44
x = 04 or x = -8 4
x = 0 or x = -2
So as we can see the roots come out to be 0 and -2. We could also use polynomial long division to find the roots of longer polynomial functions.
When trying to determine the number of complex roots (imaginary numbers) remember that complex roots always come in pairs. This will give you an indication of how many complex roots you need to find after finding the real roots. Remember the degree of the polynomial tells you the total number of roots it has! Note the example used was neither even nor odd, but the same concept applies!