A-level Mathematics/OCR/C3/Algebraic Functions

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Functions

Notation

Here is the notation that is used when dealing with mapping questions.

${\displaystyle \lbrace \ldots \rbrace }$ means a set of ...

E.g. ${\displaystyle \lbrace x:x\geq 2\rbrace }$ means a set of ${\displaystyle x}$ where ${\displaystyle x}$ is bigger than or equal to 2.

${\displaystyle f:X\rightarrow Y}$ means the function f maps the set X is mapped into the set Y. Another way to write this is ${\displaystyle f\left(x\right)=x}$.

${\displaystyle \mathbb {N} }$ - Natural Numbers – e.g. {1,2,3,. ..}

${\displaystyle \mathbb {Z} }$ - Integers – e.g. {...−2,−1,0,1,2,. ..}

${\displaystyle \mathbb {Q} }$ - Fractions (of rational numbers)

${\displaystyle \mathbb {R} }$ - Real Numbers, including rational and irrational numbers but not complex numbers such as ${\displaystyle {\sqrt {-1}}}$

Irrational numbers have no general symbol – their notation is specific - e.g. {${\displaystyle \pi \ e\ {\sqrt {x}}}$}

Definitions

Mapping: a rule connecting two sets of items.

Object/Input: an item in the starting set.

Image/Output: an item in the finishing set.

Domain: the set of all objects.

Co-domain: set of all possible finishing values.

Range: elements from the co-domain that are actually used.

Different Mappings

1) unique ${\displaystyle x}$ ${\displaystyle \rightarrow }$ unique ${\displaystyle y}$(1:1 or injective mapping)

2) several ${\displaystyle x}$ ${\displaystyle \rightarrow }$ unique ${\displaystyle y}$(many:one mapping)

3) unique ${\displaystyle x}$ ${\displaystyle \rightarrow }$ several ${\displaystyle y}$(one:many mapping)

4) several ${\displaystyle x}$ ${\displaystyle \rightarrow }$ several ${\displaystyle y}$(many:many mapping)

NB: A function can only be a 1:1 (injective) or many:one mapping.

Basic Properties

A function relates each of its inputs to exactly one output. If there are ever two outputs the formula it is not a function. For example ${\displaystyle y=x^{2}\,}$ is a function because every x has only one output, but ${\displaystyle y^{2}=x\,}$ is not a function because every x can have two values, for example 4 = ${\displaystyle -2^{2}}$ and ${\displaystyle 2^{2}}$ so x can be 2 or -2. Graphs of circles (and similar shapes) represent many:many mappings and so they are not functions. You can determine if a graph is a function by using the vertical line test. This test states that a graph is a function if and only if any vertical line drawn will intersect the graph only once. There are four ways to represent a function:

1)A Formula

2)A Graph

3)A Table

4)Verbally

Domain and Range

Every function has a domain and range. The domain is the set of the input values. The range is the set of the output values. Remember if there is a reflection or translation in the function, there will be a reflection or translation in either one or both the domain and range. In example:

The domain of ${\displaystyle f\left(x\right)=+{\sqrt {x}}}$ is ${\displaystyle \left[0,\infty \right)}$ and the range is ${\displaystyle \left[0,\infty \right)}$ but the domain of ${\displaystyle f\left(x\right)=-{\sqrt {x+6}}}$ is ${\displaystyle \left[-6,\infty \right)}$ and the range is ${\displaystyle \left[0,-\infty \right)}$. See Reflection Rule 1 and Translation Rule 2.

Domain

The domain of a function is generally all ${\displaystyle \mathbb {R} }$ with these three notable exceptions: 1) If x is in the denominator, the point at which the denominator is zero is not in the domain, E.g

${\displaystyle f(x)={\frac {x}{x-1}}\,}$ the domain will be all ${\displaystyle \mathbb {R} }$except for 1.

2) If the exponent is less than zero and an even quantity, the domain will be ${\displaystyle \left[0,\infty \right)}$, E.g

${\displaystyle {\sqrt {x}}}$ the domain will be ${\displaystyle \left[0,\infty \right)}$ but ${\displaystyle {\sqrt[{3}]{x}}}$ will be all ${\displaystyle \mathbb {R} }$.

3) In a piecewise function the domain will be restricted by the definition, E.g

${\displaystyle f(x)={\begin{cases}x,&if\ x\leq 5\\x^{2},&if\ 5 The domain of ${\displaystyle x^{2}}$ will be (5,10).

Range

There is one simple rule to remember for range. All even functions have a range of ${\displaystyle [0,\infty )}$ and odd functions have a range of all ${\displaystyle \mathbb {R} }$.

Piecewise Functions

A piecewise function is a function that is given by different expressions on different intervals. The graph of ${\displaystyle f(x)={\begin{cases}x^{2},&{\mbox{if}}\ x<2\\-\left(x-3\right)^{2}+9,&{\mbox{if}}\ x\geq 2\end{cases}}}$ is shown on the right. The open circle indicates that the point is not included on the graph and the closed circle indicates that the point is included on the graph. So ${\displaystyle f(2)}$ = 8 not 4.

One-One Function

A one-one function must have a unique output for every input. In example ${\displaystyle f(x)=x^{2}}$ will not be one-one because the inputs -2 and 2 will both produce the output 4, but ${\displaystyle f(x)=x^{3}}$ will be one-one because every input will produce a unique output, f(-2)= -8 and f(2)=8. You can determine if a graph is one-one by using the horizontal line test. This test states that a function is one-one if and only if any horizontal line drawn will intersect the graph only once.

Inverse Function

The inverse of the function will produce the input value for the output value of the function. A function must be one-one in order for the inverse of the function to exist. To illustrate this mathematically: ${\displaystyle f\left(x\right)=y}$ and ${\displaystyle f^{-1}\left(y\right)=x}$. Also the composition of the inverse and the function will produce x: ${\displaystyle ff^{-1}\left(x\right)=f^{-1}f\left(x\right)=x}$. An illustration involving apples and lemons can be found below. The original functions turns apples into lemons and the inverse function turns lemons into apples.

To find the inverse of a function.

1)Solve the equation in terms of x

2)Switch y with x.

3)Write in the notation ${\displaystyle f^{-1}\left(x\right)}$

In Example find the inverse of ${\displaystyle f\left(x\right)={\frac {x-1}{x+2}}}$.

1) ${\displaystyle y={\frac {x-1}{x+2}}\iff yx+2y=x-1\iff yx-x=-2y-1\iff x={\frac {-2y-1}{y-1}}\,}$

2)${\displaystyle y={\frac {-2x-1}{x-1}}}$

3)${\displaystyle f^{-1}\left(x\right)={\frac {-2x-1}{x-1}}}$

You can test this out now ${\displaystyle f\left(2\right)={\frac {1}{4}}}$ and ${\displaystyle f^{-1}\left({\frac {1}{4}}\right)=2}$.

Graphical Representation of One-One Functions

When we graph an inverse of a one-one function we simply have to switch the x and y co-ordinates:(x,y) becomes (y,x). The co-ordinate (5,7) will be (7,5) on the inverse graph. The graph will be a reflection through the line y=x. In the example on the right the red line is the graph of ${\displaystyle f\left(x\right)=x^{3}}$ and the green line is the graph of ${\displaystyle f^{-1}\left(x\right)={\sqrt[{3}]{x}}}$.

Composition of Functions

The mathematical symbol for composition of a function is ${\displaystyle \circ }$. Alternatively a composition can be written as ${\displaystyle f\left[g\left(x\right)\right]}$ When you compose two functions you replace all instances of the variable with the second function. You can tell the order of the function by the order that the composition is written in ${\displaystyle F\circ G}$, means F is composed of G. If the domain or range is restricted in some way in the original function it will be restricted in the same way in the composite function.In example ${\displaystyle f\left(x\right)=2x^{2}+5}$ and ${\displaystyle g\left(x\right)={\sqrt {5x}}}$ find ${\displaystyle f\circ g}$. Then find the domain and range

1) Replace all instances of x in f with ${\displaystyle {\sqrt {5x}}}$ giving ${\displaystyle f\left(x\right)=2{\sqrt {5x}}^{2}+5}$.

2) Now simplify. ${\displaystyle f\circ g\left(x\right)=10x+5}$.

3) The domain will be ${\displaystyle x\geq 0}$ because g(x) can never have a negative x value, and the range will be ${\displaystyle fg\left(x\right)\geq 0}$ because g(x) can never be negative.

This is part of the C3 (Core Mathematics 3) module of the A-level Mathematics text.