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Functions and Graphs (Metric)

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 \(g\).

The inverse of a given function.

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

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.

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

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

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

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.

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

The relationship between hyperbolic identities and trigonometric identities.