CA6 Inverse functions

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From users who are members of Skills Audits for Maths and Stats :

Lauren Desoysa	said	Ready to use	1 year ago
Megan Oliver	said	Needs to be tested	1 year, 1 month ago

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Will Morgan 11 months, 2 weeks ago

Published this.

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Lauren Desoysa 1 year ago

Gave some feedback: Ready to use

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Megan Oliver 1 year, 1 month ago

Gave some feedback: Needs to be tested

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Megan Oliver 1 year, 1 month ago

Created this as a copy of Functions: Inverse Functions 2b.

Name	Status	Author	Last Modified
Functions: Inverse Functions 1	Ready to use	Ben McGovern	10/10/2022 13:03
Functions: Inverse Functions 2a	Ready to use	Ben McGovern	10/10/2022 13:03
Functions: Inverse Functions 2b	Ready to use	Ben McGovern	10/10/2022 13:04
Functions: Inverse Functions 3	Ready to use	Ben McGovern	10/10/2022 13:04
Functions: Inverse Functions 4	Ready to use	Ben McGovern	10/10/2022 13:04
Functions: Inverse Functions 5	Ready to use	Ben McGovern	10/10/2022 13:04
CA6 Inverse functions	Ready to use	Megan Oliver	09/04/2024 15:42

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Question statement

If $f(x)=\simplify{({m}x+{c})/(x+{a})},\,x\neq \simplify{{-a}}$ , find the inverse function, $f^{-1}(x)$ .

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			Name	Type	Generated Value

m	integer	9
c	integer	3
a	integer	6

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To find $f^{-1}x$ , it can help to first set $f(x)$ to a different variable, which we will call $y$ :

$y = f(x) = \simplify{({m}x+{c})/(x+{a})}$

Since the function $f(x)$ takes us from $x$ to $y$ , the inverse function $f^{-1}$ will take us from $y$ to $x$ . So to obtain $f^{-1}$ , we want to rearrange $y=\simplify{({m}x+{c})/(x+{a})}$ so that it is $x$ as a function of $y$ :

$\begin{split} y &\,= \simplify{({m}x+{c})/(x+{a})} \\\\ \simplify{(x+{a})y} &\,= \simplify{{m}x+{c}} \\\\ \simplify{x*y+{a}y} &\,= \simplify{{m}x+{c}} \\\\ \simplify{x*y - {m}x} &\,= \simplify{{c}- {a}y} \\ \\ \simplify{x(y-{m})} &\,= \simplify{{c}-{a}y} \\\\ x&\,= \simplify{({c}-{a}y)/(y-{m})}. \end{split}$

Hence, $f^{-1}(y) =\simplify{({c}-{a}y)/(y-{m})}$ , and therefore

$f^{-1}(x) =\simplify{({c}-{a}x)/(x-{m})}.$

(Note: The inverse is valid for all values of $x$ except $x=\var{m}$ , since this would make the denominator equal to 0.)

Use this link to find resources to help you revise how to find the inverse of functions.

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Will Morgan 11 months, 2 weeks ago

Lauren Desoysa 1 year ago

Megan Oliver 1 year, 1 month ago

Megan Oliver 1 year, 1 month ago

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CA6 Inverse functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Will Morgan 11 months, 2 weeks ago

Lauren Desoysa 1 year ago

Megan Oliver 1 year, 1 month ago

Megan Oliver 1 year, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`