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find f^(-1)for the function f(x)= (1)/(x+2)

Sagot :

LammettHash

If [tex]f(x)[/tex] and [tex]f^{-1}(x)[/tex] are inverses of each other, then we have

[tex]f\left(f^{-1}(x)\right) = x[/tex]

Since [tex]f(x) = \frac1{x+2}[/tex], we have

[tex]f\left(f^{-1}(x)\right) = \dfrac1{f^{-1}(x)+2} = x[/tex]

Solve for [tex]f^{-1}(x)[/tex] :

[tex]\dfrac1{f^{-1}(x)+2} = x \\\\ 1 = x\left(f^{-1}(x)+2\right) \\\\ 1 = x f^{-1}(x) + 2x \\\\ x f^{-1}(x) = 1 - 2x \\\\ f^{-1}(x) = \dfrac{1-2x}x \\\\ \boxed{f^{-1}(x) = \dfrac1x - 2}[/tex]

(provided that x ≠ 0 and x ≠ -2)

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