The question is asking for the inverse function. To find the inverse, first switch input and output -- which is usually easiest if you use $\displaystyle y$ notation instead of $\displaystyle f(x)$. Then, solve for $\displaystyle y$.

$\displaystyle f(x)=\frac{x+3}{x-2}$

$\displaystyle y=\frac{x+3}{x-2}$

Here's where we switch:

$\displaystyle x=\frac{y+3}{y-2}$

To solve for $\displaystyle y$, we first have to get it out of the denominator. We do that by multiplying both sides by $\displaystyle (y-2)$.

$\displaystyle x(y-2)=y+3$

Distribute:

$\displaystyle xy-2x=y+3$

Get all the $\displaystyle y$ terms on the same side of the equation:

$\displaystyle xy-y=2x+3$

Factor out a $\displaystyle y$:

$\displaystyle y(x-1)=2x+3$

Divide by $\displaystyle (x-1)$:

$\displaystyle y=\frac{2x+3}{x-1}$

This is our inverse function!

$\displaystyle f^{-1}(x)=\frac{2x+3}{x-1}$

Let's say that the function $\displaystyle f(x)$ takes the input $\displaystyle a$ and yields the output $\displaystyle b$. In math terms:

$\displaystyle f(a)=b$

So, the inverse function needs to take the input $\displaystyle b$ and yield the output $\displaystyle a$:

$\displaystyle f^{-1}(b)=a$

So, to answer this question, we need to flip the inputs and outputs for $\displaystyle f(x)$. We do this by replacing $\displaystyle x$ with $\displaystyle f(x)$ (or a dummy variable; I used $\displaystyle y$) and $\displaystyle f(x)$ with $\displaystyle x$. Then we solve for $\displaystyle y$ to get our inverse function:

$\displaystyle x=(y-4)^3+3$

Now we solve for $\displaystyle y$ by subtracting $\displaystyle 3$ from both sides, taking the cube root, and then adding $\displaystyle 4$:

$\displaystyle y=\sqrt[3]{x-3}+4$

$\displaystyle y$ is our inverse function, $\displaystyle f^{-1}(x)$

The question is essentially asking this: take $\displaystyle f^{-1}(6)$ say that equals $\displaystyle a$, then take $\displaystyle g(a)$, then whatever that equals, say $\displaystyle b$, take $\displaystyle f(b)$. So, we start with $\displaystyle f^{-1}(6)$; we know that $\displaystyle f(3)=6$, so if we flip that around we know $\displaystyle f^{-1}(6)=3$. Now we have to take $\displaystyle g(3)$, but we know that is $\displaystyle 4$. Now we have to take $\displaystyle f(4)$, but we don't have that in our table; we do have $\displaystyle f^{-1}(-2)=4$, though, and if we flip it around, we get $\displaystyle f(4)=-2$, which is our answer. 

Our question is asking "What is $\displaystyle g$ of $\displaystyle f$ of $\displaystyle g$ inverse?" First we find the $\displaystyle g$ inverse of $\displaystyle 4$. Looking at the question, we see $\displaystyle g(3)=4$; if we flip that around, we get $\displaystyle g^{-1}(4)=3$. Now we need to find what $\displaystyle f(3)$ is; that is an easy one, as it is directly provided: $\displaystyle f(3)=6$. Now we need to find $\displaystyle g(6)$. Again, this isn't given, but what is given is $\displaystyle g^{-1}(2)=6$, so $\displaystyle g(6)=2$, and that is our answer. 

To find the inverse of a function, you need to change all of the $\displaystyle y$ values to $\displaystyle x$ values and all the $\displaystyle x$ values to $\displaystyle y$ values. If you flip a function over the line $\displaystyle y=x$, then you are changing all the $\displaystyle x$ values to $\displaystyle y$ values and all the $\displaystyle y$ values to $\displaystyle x$ values, giving you the inverse of your function. 

To find the inverse of a function, we need to switch all the inputs ($\displaystyle x$ variables) for all the outputs ($\displaystyle f(x)$ variables or $\displaystyle y$ variables), so if we just switch all the $\displaystyle x$ variables to $\displaystyle y$ variables and all the $\displaystyle f(x)$ variables to $\displaystyle x$ variables and solve for $\displaystyle y$, then $\displaystyle y$ will be our inverse function. 

$\displaystyle f(x)=e^{x^2-3}+\pi$

turns into the following once the variables are switched:

$\displaystyle x=e^{y^2-3}+\pi$

the first thing we do is subtract $\displaystyle \pi$ from each side; then, we take the natural log of each side. This gives us

$\displaystyle ln(x-\pi)=y^2-3$

Then we just add three to each side and take the square root of each side, making sure we have both the positive and negative roots. 

$\displaystyle \pm\sqrt{ln(x-\pi)+3}=y$

This is the inverse function of the function with which we were provided.

In order to find the inverse function, we must swap $\displaystyle x$ and $\displaystyle y$ and then solve for $\displaystyle y$.

$\displaystyle y = 4x-9$

Becomes

$\displaystyle x=4y-9$

Now we need to solve for $\displaystyle y$:

$\displaystyle x+\underset{+9}{0}=4y-\underset{+9}{9}$

Finally, we need to divide each side by 4.

$\displaystyle \frac{x+9}{4}=\frac{4y}{4}$

This gives us our inverse function:

$\displaystyle y = \frac{x+9}{4}$

To create the inverse, switch x and y making the solution   x=3y+3. 

y must be isolated to finish the problem.

Given $\displaystyle y = x^{3} - 5$

Hence $\displaystyle x^{3}=y+5$

Interchanging $\displaystyle x$ with $\displaystyle y$ we get:

$\displaystyle y^{3}=x+5$

Solving for $\displaystyle y$ results in $\displaystyle y=\sqrt[3]{x+5}$.

Interchange the $\displaystyle x$ and $\displaystyle y$ variables and solve for $\displaystyle y$.

$\displaystyle x=\frac{2}{5}y -3$

$\displaystyle x+3=\frac{2}{5}y$

$\displaystyle 5x+15=2y$

$\displaystyle y=\frac{5}{2}x+\frac{15}{2}$