How do you find the inverse of $f (x) = \frac{2 x - 3}{x + 4}$ $f(x) = (2x-3)/(x+4)$ and is it a function?

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How do you find the inverse of $f (x) = \frac{2 x - 3}{x + 4}$ $f(x) = (2x-3)/(x+4)$ and is it a function?

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Answer 1 · 2016-05-31T12:58:45

The original function is defined in $D_{f} = R - {- 4}$ $D_f=R-{-4}$ .

I rename " $f (x)$ $f(x)$ " as " $y$ $y$ " hence:

$y = \frac{2 x - 3}{x + 4}$ $y=(2x-3)/(x+4)$

Then I solve for " $x$ $x$ " :

$y \cdot (x + 4) = 2 x - 3 \Rightarrow y x - 2 x = - 3 - 4 y \Rightarrow x (y - 2) = - 3 - 4 y \Rightarrow x = \frac{- 3 - 4 y}{y - 2}$ $y*(x+4)=2x-3=>yx-2x=-3-4y=> x(y-2)=-3-4y=>x=(-3-4y)/(y-2)$

Then I switch $x$ $x$ and $y$ $y$ :

$y = \frac{- 3 - 4 x}{x - 2}$ $y=(-3-4x)/(x-2)$

And rename " $y$ $y$ " as "f-inverse" hence

$f^{- 1} (x) = \frac{- 3 - 4 x}{x - 2}$ $f^-1(x)=(-3-4x)/(x-2)$

which is defined for $D_{f^{- 1}}^{- 1} = R - {2}$ $D_(f^-1)=R-{2}$

The inverse function is a function indeed.

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How do you find the inverse of $f (x) = \frac{2 x - 3}{x + 4}$ $f(x) = (2x-3)/(x+4)$ and is it a function?

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