NSW Y11 Maths - Extension 1 Functions 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: $y = f (x) = 2 x + 1$ , by interchanging the $x$ with $y$ we have $x = 2 y + 1$ and then make $y$ the subject $y = f^{- 1} (x) = \frac{x - 1}{2}$ is its inverse.

A more complex example is $y = f (x) = x^{2}$ for $x \geq 0$ 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/

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