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

Question

asked Feb 1, 2019 171 views

1 Answer

Sasha Davydenko · Answer 1 · 2019-02-06T03:58:57+0000

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

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