Inverse Functions and Inverse Trigonometric Functions Extension Content

By the end of this unit: ● Have an understanding of inverse functions, how

These three graphs are all functions, BUT the second and third are ‘one-to-one’ functions,

The necessary and sufficient condition for the reflection of f in y=x to be

Notation: Given a function y = f(x), the inverse function is written as y

Mutually Inverse Functions A function and its inverse are mutually inverse functions. That is:

Inverse Trigonometric Functions In order for a trigonometric function to have an inverse, the

NOTE: sin-1 x is sometimes referred to as Arsin x to denote the ‘Principal’

General Solution of Trigonometric Equations

Differentiating Inverse Trigonometric Functions

Integrating Trigonometric Functions Considering the fact that integration and differentiation are opposite processes:

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Inverse Functions and Inverse Trigonometric Functions Extension Content

By the end of this unit: ● Have an understanding of inverse functions, how to find and manipulate them ● Be able to differentiate and integrate inverse functions, in particular recognise and use the standard integrals correctly. Make sure you complete some of the more challenging Fitzpatrick/Cambridge questions ● Make your summary including graphs ● attempt HSC questions on this topic under timed conditions

These three graphs are all functions, BUT the second and third are ‘one-to-one’ functions, while the first is not. A line parallel to the x-axis will cross the graph twice, so it does not have separate y elements for every x value. An example of a ‘many-to-one’ function is f(x) = sin x, whereas if you restrict the domain so f(x) = sin x between 0 ≤ x ≤ π/2 then g(x) becomes a ‘one-to-one’ restriction of f. We are interested in one-to-one functions because of the properties they have when reflected in the line y=x. What happens if you reflect the first example in y=x?

The necessary and sufficient condition for the reflection of f in y=x to be the graph of a function is that f is one-to-one. Inverse functions are formed by taking the ‘inverse operation’ or ‘undoing’ the operation of the function. however, the inverse is not always a function.

Groves

Fitzpatrick

Notation: Given a function y = f(x), the inverse function is written as y = f-1(x). Sometimes it is also written as simply f-1.

Groves

Mutually Inverse Functions A function and its inverse are mutually inverse functions. That is:

Inverse Trigonometric Functions In order for a trigonometric function to have an inverse, the function MUST have a restricted domain. For example: the inverse sine function is written as y = sin-1 x. If we restrict the curve y = sin x to a monotonic increasing curve, it will have domain -π/2 ≤ x ≤ π/2 and domain -1 ≤ y ≤ 1. This means that y = sin-1 x has domain -1 ≤ x ≤ 1 and range -π/2 ≤ y ≤ π/2. On the left is the restricted cosine curve, and y = cos-1 x on the right. What are the domain and range for each?

NOTE: sin-1 x is sometimes referred to as Arsin x to denote the ‘Principal’ value of sin-1 x, that between -π/2 ≤ y ≤ π/2.