Question

Can someone explain how to get the answer in this problem: Yes, these are separate questions!

Answer 1

Okay. First note that \(\left((f^{-1})^{-1}\right)(x)=f(x)\). (i.e. the inverse function of the inverse function is the function itself (there are some restrictions to it but you don't have to worry about it now)) Secondly, option 3 and 4 are not correct because the inverse of a straight line is a straight line. For example, option 3 is the inverse of \(f(x)=2x-3\). So it is either option 1 or 2. Try to invert both option 1 and 2 to yield the original function and see if it fits the graph.

Answer 2

I'm graphing it now

Answer 3

us that right for A

Answer 4

\[
x=\log_2(y+3)
\]
Rearrange it to y=sth.

Answer 5

what do you mean

Answer 6

Do you know how to invert a function?

Answer 7

i guess not

Answer 8

Let say you want to invert \(f^{-1}(x)=\dfrac{x-3}{2}\). First write \(y=\dfrac{x-3}{2}\). Then you swap the x and y, i.e. \(x=\dfrac{y-3}{2}\). You then rearrange the equation such that y is only on one side and x is only on the other side. That is:\[
\begin{align*}
x&=\frac{y-3}{2}\\
2x&=y-3\\
y&=2x+3
\end{align*}
\]
Try invert the third option.

Answer 9

y=2x-3

Answer 10

How about the first option?

Answer 11

umm

Answer 12

I dont know how to do that one

Answer 13

I will do the second one for you.
\[
\begin{align*}
y&=\log_2(x)+3\\
x&=\log_2(y)+3\\
x-3&=\log_2(y)\\
y&=2^{x-3}\text{ as }2^{\log_2(y)}=y.\quad \text{In fact, }n^{\log_n(x)}=x
\end{align*}
\]
Try do something similar.