Question about an inverse function

Question

I need to find the rule of the inverse of the function $f(x) = 6x + 4$.

Should I factorize is to $\frac{x}{6} - \frac{2}{3}$ or should I leave it as $\frac{x-4}{6}$.

Thanks

Answer 1

$$ y= 6x+4$$ Swap $x$ and $y$: $$x=6y+4$$Now express $y$ and you are done...

Answer 2

Both are fine, but if you want to make it obvious that it is the inverse, you can put it as $\displaystyle \frac{x-4}{6}$, since it is apparent when put next to $6x+4$.

Answer 3

Basically for inverse function you can swap $x$ and $f(x)$, which will result in $x = 6f^{-1}(x)+4 \ \implies f^{-1}(x)=\frac{x-4}{6}$ which is actually the same as $\frac{x}{6}-\frac{2}{3}$.

Note: You solved it correctly. Both ways work, but often when you work with graphs of the linear equation it is better to keep (linear) functions in a form of $ax+b$, so I would prefer having $f^{-1}(x)=\frac{1}{6} \cdot x -\frac{2}{3}$

Answer 4

The inverse function of a function " undoes" what the original function has done.

In other words, the composition of $f$ and of its inverse , $f^{-1}$ , is the identity function :

$ (f \circ f^{-1}) (x) = (f^{-1}\circ f) (x)=$ Id$(x) = x$.

Now in our case, what is the original function actually doing?

$ (x)_{(input)} \rightarrow_{\times 6 } (6x) \rightarrow_{+4 } (6x+4)_{(output)}$

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What should do a function that takes $6x+4$ as input in order to give back $x$ as output

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$(6x+4)_{(input)} \rightarrow_{-4} (6x) \rightarrow_{ /6} (x)_{(output)}$.

So, the inverse function , let's call it $g$ , first substracts $4$ from its input and then divides the difference by 6 . In other words :

$g(x) = \frac {x-4} {6}$.

Let's check. If $g$ is actually the inverse function of $f$, then $(g\circ f) (x) =x$.

$(g\circ f) (x) = g(f(x))= \frac {(6x+4) - 4} {6}= \frac {6x} {6} = x.$.

So : $g = f^{-1}$.

The composition in reverse order should also be checked.