Question

Find the inverse function of f(x) = x+2/x+1

Answer 1

Answer:

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

Step-by-step explanation:

Let y = f(x) and rearrange making x the subject

y = $\frac{x+2}{x+1}$ ( multiply both sides by (x + 1)

y(x + 1) = x + 2 ← distribute left side

yx + y = x + 2 ( subtract x from both sides )

yx - x + y = 2 ( subtract y from both sides )

yx - x = 2 - y ← factor out x on the left side

x(y - 1) = 2 - y ← divide both sides by (y - 1 )

x = $\frac{2-y}{y-1}$

Change y back into terms of x, thus

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

Answer 2

Answer:   $\bold{f^{-1}(x)=\dfrac{2-x}{x-1}\qquad \implies \qquad f^{-1}(x)=-\dfrac{x-2}{x-1}}$

Step-by-step explanation:

Inverse is when you swap the x's and y's and solve for y.  f(x) is y.

$y=\dfrac{x+2}{x+1}\\\\\\\\\underline{\text{Swap the x's and y's}}\\\\x=\dfrac{y+2}{y+1}\\\\\\\\\underline{\text{solve for y}}\\\\x(y+1)=y+2\\\\xy+x=y+2\qquad \qquad \text{distributed x into y+1}\\\\xy-y=2-x\qquad \qquad \text{subtracted x and y from both sides}\\\\y(x-1)=2-x\qquad \qquad \text{factored out y on the left side}\\\\\\y=\dfrac{2-x}{x-1}}\qquad \qquad \text{divided both sides by x-1}$

$\large\boxed{f^{-1}(x)=\dfrac{2-x}{x-1}}$

This is equal to: $f^{-1}(x)=-\dfrac{x-2}{x-1}$