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Let k(w)= w+3/w+9. Find k^-1(-4)

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Answer:


\displaystyle k^(-1) (-4) = -(39)/(5)

Explanation:

We are given the function:


\displaystyle k(w) = (w + 3)/(w + 9)

And we want to find k⁻¹(-4).

Recall that by the definition of inverse functions:


\displaystyle \text{If } f(a) = b, \text{ then } f^(-1)(b) = a

Let k⁻¹(-4) = x, where x is an unknown value. Then by definition, k(x) must equal -4.

So:


\displaystyle k(x) = (x+3)/(x+9) = -4

Solve for x:


\displaystyle \begin{aligned} (x+3)/(x+9) &= -4 \\ \\ x+3 &= -4(x+9) \\ \\ x+3 &= -4x - 36 \\ \\ x &= -(39)/(5) \end{aligned}

Hence, k(-39/5) = -4. By definition of inverse functions, then, k⁻¹(-4) = -39/5.

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