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Revision:OCR Core 3 - FunctionsTSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > OCR Core 3 - Functions
2. FunctionsThe domain of a functionSome functions cannot be represented in the sense that it is not possible to show every value for which the function is defined. The graph of Sometimes values for which a function is undefined (mathematically) are not the only restrictions. Consider a function which calculates the volume of a cube, of side length "x" ( The domain of a function is simply the set of numbers for which that function is defined (and applicable). Hence, the function of "x" The range of a functionAs the values which can be input into a function can be restricted, it is interesting to consider how the value that the function can take (or output) can be limited. Functions such as Example 1. Calculate the domain and range of the function of "x": The range for this function is all real "x" ( This question can be approached using differential calculus. Consider that this is a quadratic function whose highest power term has a positive coefficient, meaning that there would be a minimum value if it were to be plotted. The reason behind considering the function as a graph is simply that this allows the representation of a large number of values of "x" in an intuitive manner.
Hence the minimum is at: Calculate the corresponding value of the function: Hence the range is: Function notationFunctions are like a series of operations, and as such they can be represented in a flow diagram, with arrows linking the different operations of the function. These diagrams are useful for finding the inverse function (later in this section of the notes), but are generally rather large and cumbersome. Instead, the following notation is used:
This shows that the function "f" maps all "x" in the domain onto a value of "f(x)" in the range. For example: Forming composite functionsThe idea of the flow diagram can be brought into the functions. For instance, if
Within this function there are several other functions, "add 2", "square", "add 4"; this means that the function "f" is called a composite of the aforementioned functions, with the components being the aforementioned functions. This can be shown more generally:
If the brackets (on the end expression) are removed, the remaining expression can be used, however this shows that the most inner function ( Domain and range revisitedIn Core 3 the functions used are defined only by real numbers, however it has already been demonstrated that the numbers for which a function is defined will be limited (for some functions). The natural domain refers to the numbers for which the function will give an answer (so for the example of restrictions based upon the subject of the question the natural domain is not being used). In some cases, a composite function will not give defined values for numbers which are in the domain and range of the components:
Both of the components in the above function are defined for Consider a general composite function (with two components):
This implies that the output of the function "f" is the input of the function "g", and therefore the range of "f" must be a subset of the domaim of "g" in order for all values in the domain of "f" to produce defined values by "gf". In the previous general example, the problem lies in that the domain of Reversing functionsMost people are aware of inverse operations, however this idea can be extended to a series of operations, and therefore functions. The inverse function (with respect to some function) is merely a function such that all values in the domain are mapped onto the values which produced them by the original function. Consider the easy example:
This function has a single operation, so the inverse is easily defined as:
So, in general, the inverse function is a function Note that some functions will invert themselves, such as "divide by -1". One-one functionsFor some functions there are several values which are mapped onto the same value by the function; the trigonometric functions are good examples of these. These functions are not one-one functions with their natural domain as there is more than one value that will map onto a given value. As such, inverse functions are not defined for those functions that are not one-one (as there would be ambiguity as to which value would be meant; functions could have their inverses applied and produce different answers [against the idea of an inverse]). The identity functions are Finding inverse functionsIn the previous example, finding the inverse was very easy, as it is simple to see that the single operation is inverted by using the inverse operation. In a more complex situation a flow diagram might be of use, for example:
First this can be represented as a flow diagram:
It will probably be obvious that working backwards will produce the inverse function:
It is simple to explain why this works in a more general sense. For functions which can be defined in this manner, there is simply a series of operations being carried out; consider these as functions, hence:
This implies that the first function applied was Another method is useful in cases where it is not possible to use this flow diagram method.
Therefore:
Therefore it is sufficient to find some function of "y" which maps onto "x". Example 1. Find
Hence:
(Note that the letter used does not matter, and often it is more understandable to use the letter used in the original question to defined the inverse function [it is also more economical in the context of use of letters]). In some cases where a quadratic is solved (in order to obtain the inverse function) the plus or minus symbol must be dealt with. Whether the plus or the minus is chosen will depend on the original function. Graphing inverse functionsConsider a graph of The following (equivalent) equation can be formed:
Reflecting this in the line
Also SeeRead these other OCR Core 3 notes:
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![8 \rightarrow [add \ 2] \rightarrow 10 \rightarrow [square] \rightarrow 100 \rightarrow [add \ 4] \rightarrow 104](http://www.thestudentroom.co.uk/latexrender/pictures/25/257627ac8dc7886cdd33e65c870d8240.png "8 \rightarrow [add \ 2] \rightarrow 10 \rightarrow [square] \rightarrow 100 \rightarrow [add \ 4] \rightarrow 104")
![x \rightarrow [f_{1}] \rightarrow f_{1}(x) \rightarrow [f_{2}] \rightarrow f_{2}(f_{1}(x)) \rightarrow ... \rightarrow [f_{n}] \rightarrow f_{n}(f_{n - 1}( ... f_{2}(f_{1}(x)))...))](http://www.thestudentroom.co.uk/latexrender/pictures/f8/f87438bb43e5794a83cb72a6b92fc167.png "x \rightarrow [f_{1}] \rightarrow f_{1}(x) \rightarrow [f_{2}] \rightarrow f_{2}(f_{1}(x)) \rightarrow ... \rightarrow [f_{n}] \rightarrow f_{n}(f_{n - 1}( ... f_{2}(f_{1}(x)))...))")
![x \rightarrow [double] \rightarrow 2x \rightarrow [add \ 3] \rightarrow 2x + 3](http://www.thestudentroom.co.uk/latexrender/pictures/59/595ab781cc918a541f3b805b6cea214c.png "x \rightarrow [double] \rightarrow 2x \rightarrow [add \ 3] \rightarrow 2x + 3")
![2x + 3 \rightarrow [subtract \ 3] \rightarrow 2x \rightarrow [halve] \rightarrow x](http://www.thestudentroom.co.uk/latexrender/pictures/e1/e1d0fe0df2f36c6ddbf02e5afda8c77f.png "2x + 3 \rightarrow [subtract \ 3] \rightarrow 2x \rightarrow [halve] \rightarrow x")
