Question

Please help me solve this problem step by step. I don’t get it

Answer 1

Part a

we have the function

$f(t)=\log_2(t)$

Find out the inverse

step 1

Let

y=f(t)

$y=\operatorname{\log}_2(t)$

step 2

Exchange the variables (y for t and t for y)

$t=\log_2y$

step 3

Isolate the variable y

Apply the definition of logarithm

$\begin{gathered} 2^t=y \\ y=2^t \\ f^{-1}(t)=2^t \end{gathered}$

Part B

we have the function

$g(t)=\frac{1}{t-2}$

Find out the inverse of function g(t)

$y=\frac{1}{t-2}$

Exchange the variables

$t=\frac{1}{y-2}$

Isolate the variable y

$\begin{gathered} t(y-2)=1 \\ y-2=\frac{1}{t} \\ y=\frac{1}{t}+2 \\ \\ g^{-1}(t)=\frac{1}{t}+2 \end{gathered}$

The domain of the inverse function is all real numbers except for t=0

so

interval (-infinite,0) U (0, infinite)