NSW Y11 Maths - Extension 1 Functions Inverse Functions

Resources for Inverse Functions

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Inverse Functions Theory

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  • Inverse Functions - Video - How To Find The Inverse of a Function

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Theory

In general, if \(y = f(x)\) is a function, then for each \(x\) in the domain, there is one and only one value of \(y\). The relation obtained by interchanging \(x\) and \(y\) is \(x = f(y)\). If for each \(x\) in the domain, \(y\) is uniquely determined, then this represents a new function, called the inverse function to \(y = f(x)\) and is denoted by \(y = {f^{ - 1}}(x)\).

To obtain the inverse function \(y = {f^{ - 1}}(x)\) from \(y = f(x)\), \(x\) and \(y\) are interchanged and consequently, so are the domains and ranges.

The domain of \(y = {f^{ - 1}}(x)\) is the range of \(y = f(x)\) and the range of \(y = {f^{ - 1}}(x)\) is the domain of \(y = f(x)\).

An example is: \(y = f(x) = 2x + 1\), by interchanging the \(x\) with \(y\) we have \(x = 2y + 1\) and then make \(y\) the subject \(y = {f^{ - 1}}(x) = \dfrac{{x - 1}}{2}\) is its inverse.

A more complex example is \(y = f(x) = {x^2}\) for \(x \ge 0\) here \(x = {y^2} \to y = \pm \sqrt x \)

Remember the domain of \(y = f(x)\) is the range of the inverse hence \({f^{ - 1}}(x) = \sqrt x \).

Syllabus Reference

NSW Syllabus Reference: ME-F1.3: Inverse functions. This will require student to 

  • define the inverse relation of a function \(y=f(x)\) to be the relation obtained by reversing all the ordered pairs of the function
  • examine and use the reflection property of the graph of a function and the graph of its inverse (ACMSM096)
  • write the rule or rules for the inverse relation by exchanging \(x\) and \(y\) in the function rules, including any restrictions, and solve for \(y\), if possible
  • when the inverse relation is a function, use the notation \(f^{-1}(x)\) and identify the relationships between the domains and ranges of \(f(x)\) and \(f^{-1}(x)\)
  • when the inverse relation is not a function, restrict the domain to obtain new functions that are one-to-one, and compare the effectiveness of different restrictions
  • solve problems based on the relationship between a function and its inverse function using algebraic or graphical techniques

Ref: https://educationstandards.nsw.edu.au/