# vol2sample1 - Te A Le l Maths ach ve Vol. 2: A2 C Module...

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Demo Disc Demo Disc Teach A Level Maths” Vol. 2: A2 Core Modules © Christine Crisp

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47: Solving Differential Equations 28: Integration giving Logs 31: Double Angle Formulae 18: Iteration Diagrams and Convergence 15: More Transformations 13: Inverse Trig Ratios 3: Graphs of Inverse Functions 2: Inverse Functions The slides that follow are samples from the 55 presentations that make up the work for the A2 core modules C3 and C4. 16: The Modulus Function 22: Integrating the Simple Functions 4: The Function x e y = 43: Partial Fractions 51: The Vector Equation of a Line
Explanation of Clip-art images An important result, example or summary that students might want to note. It would be a good idea for students to check they can use their calculators correctly to get the result shown. An exercise for students to do without help.

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2: Inverse Functions Demo version note: The students have already met the formal definition of a function and the ideas of domain and range. In the following slides we prepare to introduce the condition for an inverse function.
x y sin = 1 3 + = x y One-to-one and many-to-one functions Each value of x maps to only one value of y . . . Consider the following graphs Each value of x maps to only one value of y . . . BUT many other x values map to that y. and each y is mapped from only one x. and Inverse Functions

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One-to-one and many-to-one functions is an example of a one-to-one function 1 3 + = x y is an example of a many-to-one function x y sin = x y sin = 1 3 + = x y Consider the following graphs and Inverse Functions
3: Graphs of Inverse Functions Demo version note: By this time the students know how to find an inverse function. The graphical link between a function and its inverse has also been established and this example follows.

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e.g. On the same axes, sketch the graph of 2 , ) 2 ( 2 - = x x y N.B! ) 0 , 2 ( x y = ) 4 , 4 ( x Solution: ) 2 , 0 ( ) 3 , 1 ( Graphs of Inverse Functions
e.g. On the same axes, sketch the graph of and its inverse. 2 , ) 2 ( 2 - = x x y N.B! x y = 2 ) 2 ( - = x y Solution: N.B. Using the translation of we can see the inverse function is . x 2 ) ( 1 + = - x x f 2 + = x y Graphs of Inverse Functions

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2 ) 2 ( - = x y 2 + = x y A bit more on domain and range The domain of is . 2 x ) ( x f Since is found by swapping x and y , ) ( 1 x f - the values of the domain of give the values of the range of . ) ( x f ) ( 1 x f - 2 ) 2 ( ) ( - = x x f 2 x Domain 2 y 2 ) ( 1 + = - x x f Range 2 , ) 2 ( ) ( 2 - = x x x f The previous example used . Graphs of Inverse Functions
2 ) 2 ( - = x y A bit more on domain and range 2 , ) 2 ( ) ( 2 - = x x x f The previous example used . The domain of is . 2 x ) ( x f Similarly, the values of the range of ) ( x f ) ( 1 x f - give the values of the domain of 2 + = x y Since is found by swapping x and y , ) ( 1 x f - Graphs of Inverse Functions the values of the domain of give the values of the range of . ) ( x f ) ( 1 x f -

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Demo version note: The exponential function has been defined and here we build on earlier work to find the inverse and its graph.
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## This note was uploaded on 01/12/2011 for the course MAT117 MAT 117 taught by Professor Ranjitrebello during the Spring '09 term at University of Phoenix.

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vol2sample1 - Te A Le l Maths ach ve Vol. 2: A2 C Module...

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