Question

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

Answer 1

Answer:

$\displaystyle k^{-1} (-4) = -\frac{39}{5}$

Step-by-step explanation:

We are given the function:

$\displaystyle k(w) = \frac{w + 3}{w + 9}$

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:

$\displaystyle k(x) = \frac{x+3}{x+9} = -4$

Solve for x:

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

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