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Functions and Graphs (Metric)

Class Details

Tom Baker

This class covers the idea of a function, composites and inverses, odd and even functions and specific functions including the modulus function, the exponential and logarithmic functions and the hyperbolic functions. Some of this is covered at A-level, and most of the rest is Further Maths.

Functions in General

Composites, inverses, odd and even functions. This is an A-level topic, which Science and Engineering courses at Imperial will assume you know.

Functions of the form \(f(g(x))\) for given \(f\) and \(g\).

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The inverse of a given function.

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Odd and even functions; the odd and even parts of a function.

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Particular Functions

Modulus, Heaviside step, exponential and logarithmic functions. Apart from the Heaviside function, which is a university topic, this is covered at A-level and Science and Engineering courses will assume you know it.

The function \(y=|x|\); composition with other functions; solving equations and inequalities.

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The Heaviside step function \(H(x)\), and its use in characterising piecewise functions.

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Exponential functions in general, and the exponential function, \(y=e^x\); laws of exponents, and simplifying expressions involving exponentials.

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Logarithmic functions in general, and the natural logarithm; laws of logarithms and the change of base theorem. Simplifying expressions.

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Hyperbolic Functions

The functions \(\sinh x\), \(\cosh x\) and \(\tanh x\), and their reciprocals and inverses. This is a Further Maths topic; Imperial courses that require Further Maths may assume you know it.

The functions \(\sinh x\), \(\cosh x\) and \(\tanh x\) and their reciprocals. Solving equations involving them.

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The inverse functions \(\sinh^{-1}\,x\), \(\cosh^{-1}\,x\) and \(\tanh^{-1}\,x\); logarithmic form for each.

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The relationship between hyperbolic identities and trigonometric identities.

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