Mastering Mathematics Smartly by Wee Wen Shih

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Inverse functions

Let us summarise some key points:

(1) For the inverse of a function f to exist, check that f is one-one from its graph.

(2) To find the rule of the inverse of f, let y = f (x). Then express x in terms of y. If x has 2 expressions, use the domain of f to decide which one is appropriate.

(3) The domain of the inverse of f is the range of f.

(4) The range of the inverse of f is the domain of f.

(5) The graphs of the inverse of f and of f are reflections in the line y = x.

See example.

Try this: Mathematics 9233 Nov '05/Paper 1/Q11

Note: Given y = \text{f}(x), an expression for \text{f}^{-1}(x) may not always be found by algebraic means, e.g. y = x\:\text{ln}\:x. However one can still sketch the graph of the inverse of y = x\:\text{ln}\:x, via reflection about the line y = x.

Modulus functions

Modulus functions often pose a great difficulty to students.

Take for instance, y = \text{f}(x) = \text{e}^{|x|} where x \in \mathbb{R}.

The method to deal with the modulus sign is to apply its definition:
|x| = x if x \geq 0 
or
|x| = -x if x < 0.

With this definition, we see that our original function is composed of two parts:
y = \text{e}^x if x \geq 0 
and
y = \text{e}^{-x} if x < 0.

If we proceed to sketch this graph, we will see that the function is not one-one (via the horizontal line test) and hence, its inverse will not exist.

It is common to restrict (i.e. make smaller) the original domain (i.e. x \in \mathbb{R}), so that the inverse can be made to exist. Two possibilities are:
x \geq 0
and
x \leq 0.

If we consider the maximal domain x \geq 0, we shall take y = \text{e}^x and then proceed to express x in terms of y to find the inverse. As a result, we will obtain \text{f}^{-1}(x) = \text{ln}\:x.

On the other hand when we consider the maximal domain x \leq 0, we shall take y = \text{e}^{-x} and then go on to express x in terms of y to find the inverse. As a result, we will obtain \text{f}^{-1}(x) = -\text{ln}\:x.

In summary, one must be careful to select the correct part of the modulus function based on the restricted domain, so that the correct inverse may be found.