Examples of state function in the following topics:
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- The two-chamber structure had functioned well in state governments.
- Two senators were chosen by state governments which benefited smaller states.
- When the Constitution was ratified in 1787, the ratio of the populations of large states to small states was roughly 12 to one.
- Since each state has two senators, residents of smaller states have more clout in the Senate than residents of larger states.
- Critics, such as constitutional scholar Sanford Levinson, have suggested that the population disparity works against residents of large states and causes a steady redistribution of resources from large states to small states.
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- An inverse function is a function that undoes another function.
- Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function.
- Not all functions have an inverse.
- Let's take the function $y=x^2+2$.
- We can check to see if this inverse "undoes" the original function by plugging that function in for $x$:
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- A wave function is a probability amplitude in quantum mechanics that describes the quantum state of a particle and how it behaves.
- In quantum mechanics, a wave function is a probability amplitude describing the quantum state of a particle and how it behaves.
- For a single particle, it is a function of space and time.
- The most common symbols for a wave function are ψ(x) or Ψ(x) (lowercase or uppercase psi, respectively), when the wave function is given as a function of position x.
- The trajectories C-F are examples of standing waves, or "stationary states. " Each standing-wave frequency is proportional to a possible energy level of the oscillator.
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- A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.
- One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.
- Notice also, that these two ordered pairs form a horizontal line; which also means that the function is not one-to-one as stated earlier.
- The graph of the function $f(x)=x^2$ fails the horizontal line test and is therefore NOT a one-to-one function.
- If a horizontal line can go through two or more points on the function's graph then the function is NOT one-to-one.
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- Although nitriles do not have a carbonyl group, they are included here because the functional carbon atoms all have the same oxidation state.
- Functional groups of this kind are found in many kinds of natural products.
- Some examples are shown below with the functional group colored red.
- Most of the functions are amides or esters, cantharidin being a rare example of a natural anhydride.
- Penicillin G has two amide functions, one of which is a β-lactam.
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- Powers of local governments are defined by state rather than federal law, and states have adopted a variety of systems of local government.
- Local government in the United States is structured in accordance with the laws of the individual states, territories and the District of Columbia.
- A number of independent cities operate under a municipal government that serves the functions of both city and county.
- In particular, towns in New England have considerably more power than most townships elsewhere and often function as independent cities in all but name, typically exercising the full range of powers that are divided between counties, townships and cities in other states.
- In some states, a city can become independent of any separately functioning county government and function both as a county and as a city.
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- A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
- Otherwise, a function is said to be a "discontinuous function."
- A continuous function with a continuous inverse function is called "bicontinuous."
- This function is continuous.
- In fact, a dictum of classical physics states that in nature everything is continuous.
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- Function notation, $f(x)$ is read as "$f$ of $x$" which means "the value of the function at $x$."
- The ordered pairs normally stated in linear equations as $(x,y)$, in function notation are now written as $(x,f(x))$.
- The graph for this function is below.
- The degree of the function is 3, therefore it is a cubic function and is sometimes shaped like the letter N.
- The function is linear, since the highest degree in the function is a $1$.
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- Each trigonometric function has an inverse function that can be graphed.
- In order to use inverse trigonometric functions, we need to understand
that an inverse trigonometric function “undoes” what the original
trigonometric function “does,” as is the case with any other function
and its inverse.
- However, the sine, cosine, and tangent functions are not
one-to-one functions.
- As with other
functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.
- In summary, given the above reasoning and graphs of the inverse trigonometric functions above, we can state the following relations:
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- A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
- Because of this, the discussion of organic reactions is often organized according to functional groups.
- The following table summarizes the general chemical behavior of the common functional groups.
- For reference, the alkanes provide a background of behavior in the absence of more localized functional groups.
- For example, addition reactions to C=C are significantly different from additions to C=O, and substitution reactions of C-X proceed in very different ways, depending on the hybridization state of carbon.