Question

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

Answer 1

`\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.