Question

Can anyone help out with question 5a and 5b?

Answer 1

To find the inverse, let's isolate x:

`y=k\log_2(x) \iff (y)/(k) = \log_2(x) \iff 2^{(y)/(k)} = 2^(\log_2(x)) \iff 2^{(y)/(k)} = x`

So, the inverse function is

`f^(-1)(x) = 2^{(x)/(k)}`

We know that the inverse gives 8 when evaluated at x=1, so

`f^(-1)(1) = 2^{(1)/(k)} =8`

Now, since
`8=2^3`, we need

`(1)/(k)=3 \iff k=(1)/(3)`

So, the inverse function is

`f^(-1)(x) = 2^(3x)`

Which means that

`f^(-1)\left((2)/(3)\right) = 2^{3(2)/(3)}=2^2=4`