Examples of odd function in the following topics:
-
- We have previously discussed even and odd functions.
- Recall that even functions are symmetric about the $y$-axis, and odd functions are symmetric about the origin, $(0, 0)$.
- On the other hand, sine and tangent are odd functions because they are symmetric about the origin.
- Graphs that are symmetric about the origin represent odd functions.
- For odd functions, any two points with opposite $x$-values also have opposite $y$-values.
-
- Functions can be classified as "odd" or "even" based on their composition.
- The terms "odd" and "even" can only be applied to a limited set of functions.
- Oftentimes, the parity of a function will reveal whether it is odd or even.
- How can we check if a function is odd or even?
- The function, $f(x)=x^3-4x$ is odd since the graph is symmetric about the origin.
-
- A periodic function is a function with a repeated set of values at regular intervals.
- Specifically, it is a function for which a specific horizontal shift, $P$, results in a function equal to the original function:
- As we can see in the graph of the sine function, it is symmetric about the origin, which indicates that it is an odd function.
- This is characteristic of an odd function: two inputs that are opposites have outputs that are also opposites.
- The sine function is odd, meaning it is symmetric about the origin.
-
- Suppose now we consider an odd function (i.e., $f(x) = -f(-x)$ ), such as $f(x) = x$ .
- So why the odd behavior at the endpoints?
- It's because we've assume the function is periodic on the interval $[-1,1]$.
- In other words any non-periodic function defined on a finite interval can be used to generate a periodic function just by cloning the function over and over again.
- Figure 4.4: First four nonzero terms of the Fourier series of the function f(x) = abs(x).
-
- Consider the polynomial function:
- If $n$ is odd and $a_n$ is positive, the function declines to the left and inclines to the right.
- If $n$ is odd and $a_n$ is negative, the function inclines to the left and declines to the right.
- Except when $x$ is negative and $n$ is odd; then the opposite is true.
- Because the degree is odd and the leading coefficient is positive, the function declines to the left and inclines to the right.
-
- Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics.
- For example, if a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of the quartic function by reducing it into a quadratic form.
- Consider a quadratic function with no odd-degree terms which has the form:
-
- Symptoms for ODD are of three types: angry/irritable mood, argumentative/defiant behavior, and vindictiveness.
- For a child or adolescent to qualify for a diagnosis of ODD, behaviors must cause considerable distress for the family or interfere significantly with academic or social functioning.
- Several preventative programs have had a positive effect on those at high risk for ODD.
- The outbursts cannot be premeditated and must cause distress or impairment of functioning, or lead to financial or legal consequences.
- Lesions in these areas are also associated with improper blood sugar control, leading to decreased brain function in these areas, which are associated with planning and decision making.
-
- Some projects use the parity of the minor number component to indicate the stability of the software: even means stable, odd means unstable.
- The advantage of the even/odd system, which has been used by the Linux kernel project among others, is that it offers a way to release new functionality for testing without subjecting production users to potentially unstable code.
- The development team handles the bug reports that come in from the unstable (odd-minor-numbered) series, and when things start to settle down after some number of micro releases in that series, they increment the minor number (thus making it even), reset the micro number back to "0", and release a presumably stable package.
- The even/odd system is probably best for projects that have very long release cycles, and which by their nature have a high proportion of conservative users who value stability above new features.
- It is not the only way to get new functionality tested in the wild, however. the section called "Stabilizing a Release" describes another, perhaps more common, method of releasing potentially unstable code to the public, marked so that people have an idea of the risk/benefit trade-offs immediately on seeing the release's name.
-
- For example, consider the analysis of proportions in the case study "Mediterranean Diet and Health. " In this study, one group of people followed the diet recommended by the American Heart Association (AHA), whereas a second group followed the "Mediterranean Diet. " One interesting comparison is between the proportions of people who were healthy throughout the study as a function of diet.
- A third commonly used measure is the "odds ratio. " For our example, the odds of being healthy on the Mediterranean diet are 90:10 = 9:1; the odds on the AHA diet are 79:21 = 3.76:1.
- The ratio of these two odds is 9/3.76 = 2.39.
- Therefore, the odds of being healthy on the Mediterranean diet is 2.39 times the odds of being healthy on the AHA diet.
- Note that the odds ratio is the ratio of the odds and not the ratio of the probabilities.
-
- An odds ratio is the ratio of two odds.
- In order to compute the odds ratio, one follows three steps:
- Divide the first odds by the second odds to obtain the odds ratio.
- The odds of a man drinking wine are $90$ to $10$ (or $9:1$) while the odds of a woman drinking wine are only $20$ to $80$ (or $1:4=0.25:1$).
- The log odds ratio shown here is based on the odds for the event occurring in group $B$ relative to the odds for the event occurring in group $A$.