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Discover how to solve inverse functions both graphically and algebraically, whilst investigating their relations, properties and patterns.

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Introduction

Year 11 IB Maths HL – Inverse FunctionsMathematical Investigation

Graphical Determination of the Inverse                                                                         Fergal Banks

Problem Statement: Whilst completing this investigation, I plan to find out three main aspects about inverse functions,

  • Can we find the inverse of any given function graphically?
  • What are the properties of the inverses of some common functions?
  • Do all Functions have an inverse?

Whilst using the method, the function f(x) has an inverse f¯¹ (x) if f (f¯¹ (x))  =  x.

Method: Discover how to solve

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Middle

(3x - 1)/(x + 2) is reflected and becomes, (2x + 1)/(3 – x). This results in a mirror image of the original function.

c)        image02.png

f(x) = x³ is reflected resulting in the inverse function, g(x) = ³√x.

image03.png

3.

Using the linear function, f(x) = 4x + 8, it is clear that my results in Q2 are indeed correct, as they are confirmed by the inverse function of the above linear function. It is flipped resulting in g(x) =. I worked out the inverse function by working

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Conclusion

Algebraically, this can proven by showing that the inverse of y = x², is y = √x. This is due to the fact that the inverse function is in fact a mirror image of the f(x). However, as each y value has more than one x value, it cannot be a function.

Conclusion: By completing this investigation, I have been able to find out that all linear and rational functions have an inverse. However, the same cannot be said about quadratic functions because it does not fulfil the criterion of a inverse function, that each y value cannot have more than one x value. All functions that complete the vertical line test will have an inverse function.

...read more.

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