Finding the inverse for a composite function

Many students know how to find the inverse of a regular function. We simply let $f(x) = y$ then proceed to make $x$ the subject. Some might not know that if you want to find the inverse of the composite function, knowing that $(fg)^{-1} = g^{-1}f^{-1}$ will be useful. Do note the switch in position of the functions.

At times, it might be really impossible to make out the inverse of $fg(x)$ by doing the regular $fg(x)=y$ and then proceeding to make $x$ the subject. Thus, this formula will come in handy.

The following is a simple proof. Assuming all the following composite functions exists.

Let $\phi(x)= (fg)^{-1}(x)$

Then $\phi^{-1}(x) = fg(x)$
$f^{-1} [\phi^{-1}(x)] = f^{-1}[fg(x)] = g(x)$

$g^{-1} \{f^{-1} [\phi^{-1}(x)]\} = g^{-1}g(x) = x$

We have that $g^{-1} f^{-1} \phi^{-1}(x) = x$

Considering $g^{-1} f^{-1} \phi^{-1}[\phi (x)] = \phi(x)$

$\therefore, g^{-1} f^{-1} (x) = \phi(x)$

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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Finding the inverse for a composite function

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