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**“Teach A Level Maths” Vol. 2: A2 Core Modules**

1: Functions

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**A function is a rule , which calculates values of**

for a set of values of x. e.g and are functions. is called the image of x Another Notation means is often replaced by y.

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**x x We can illustrate a function with a diagram . -1 -3 . -1 . 1.5 2 .**

. 2.1 1.5 -3 . x x -1 . 2 . A few of the possible values of x 3.2 . The rule is sometimes called a mapping.

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A bit more jargon To define a function fully, we need to know the values of x that can be used. The set of values of x for which the function is defined is called the domain. In the function any value can be substituted for x, so the domain consists of all real values of x We write R We say “ real ” values because there is a branch of mathematics which deals with numbers that are not real. stands for the set of all real numbers R means “ belongs to the set ” So, means x belongs to the set of real numbers R

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What is a real number? The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as p and the 2; or, a real number can be given by an infinite decimal representation, such as , where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line. Real numbers measure continuous quantities such as weight, height, speed, time.

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**The range of a function is the set of values given by .**

e.g. Any value of x substituted into gives a positive ( or zero ) value. So the range of is If , the range consists of the set of y-values, so domain: x-values range: y-values Tip: To help remember which is the domain and which the range, notice that d comes before r in the alphabet and x comes before y.

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**The set of values of x for which the function is defined is called the domain.**

The range of a function is the set of values given by the rule. domain: x-values range: y-values

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**e.g. 1 Sketch the function where**

and write down its domain and range. Solution: The quickest way to sketch this quadratic function is to find its vertex by differentiating. so the vertex is

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**BUT we could have drawn the sketch for any values of x.**

So, the graph of is x domain: The x-values on the part of the graph we’ve sketched go from -5 to BUT we could have drawn the sketch for any values of x. So, we get ( x is any real number ) R BUT there are no y-values less than -5, . . . so the range is ( y is any real number greater than, or equal to, -5 )

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**) ( x f y = e.g.2 Sketch the function where .**

Hence find the domain and range of Solution: is a translation from of ) ( x f y = so the graph is: domain: x-values range: y-values ( We could write instead of y )

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Asymptotes: if a function approaches a particular x or y value then the equation of this value is called an asymptote. Notice how each value of x maps (links) to ONLY one value of y . This is the definition of a function. If each value of x maps to MORE than one value of y it is NOT a function

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**Vertical asymptote at x = 2 **

As you cannot divide by zero x – 2 0 Domain : x 2 Vertical asymptote at x = 2 The function approaches the line x = 2 but never meets it Horizontal asymptote at y = 3. The graph approaches the line y = 3 but never meets it. Range : y 3 or y<3 and y>3

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SUMMARY To define a function we need a rule and a set of values. Notation: means For , the x-values form the domain the or y-values form the range e.g. For , the domain is the range is or R

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Exercise 1. Sketch the functions where For each function write down the domain and range Solution: (a) (b) domain: R domain: R range: R range:

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**x + 3 must be greater than or equal to zero.**

We can sometimes spot the domain and range of a function without a sketch. e.g. For we notice that we can’t square root a negative number ( at least not if we want a real number answer ) so, x + 3 must be greater than or equal to zero. So, the domain is The smallest value of is zero. Other values are greater than zero. So, the range is

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**One-to-one and many-to-one functions**

Consider the following graphs and Each value of x maps to only one value of y . . . Each value of x maps to only one value of y . . . and each y is mapped from only one x. BUT many other x values map to that y.

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**One-to-one and many-to-one functions**

Consider the following graphs and is an example of a many-to-one function is an example of a one-to-one function

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**Other one-to-one functions are:**

and

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**Other many-to-one functions are:**

Here the many-to-one function is two-to-one ( except at one point ! ) This is a many-to-one function even though it is one-to-one in some parts. It’s always called many-to-one.

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**We’ve had one-to-one and many-to-one functions, so what about one-to-many?**

One-to-many relationships do exist BUT, by definition, these are not functions. e.g. is one-to-many since it gives 2 values of y for all x values greater than 1. This is not a function. Functions cannot be one-to-many. So, for a function, we are sure of the y-value for each value of x. Here we are not sure.

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