Let's solve the problem step by step from the image.

Given Functions:

• $f(x) = x + 3$
• $g(x) = 3x - 5$

We are asked to determine the following:

1. $f^{-1}(x)$ and $g^{-1}(x)$ (inverse functions).
2. The composition $(f^{-1} \circ g^{-1})(x)$.
3. The composition $(g^{-1} \circ f^{-1})(x)$.

Step 1: Finding the inverse of $f(x)$

The function is $f(x) = x + 3$. To find $f^{-1}(x)$:

1. Set $y = f(x) = x + 3$.
2. Solve for $x$: $y = x + 3 \quad \Rightarrow \quad x = y - 3.$ So, the inverse function is: $f^{-1}(x) = x - 3.$

Step 2: Finding the inverse of $g(x)$

The function is $g(x) = 3x - 5$. To find $g^{-1}(x)$:

1. Set $y = g(x) = 3x - 5$.
2. Solve for $x$: $y = 3x - 5 \quad \Rightarrow \quad 3x = y + 5 \quad \Rightarrow \quad x = \frac{y + 5}{3}.$ So, the inverse function is: $g^{-1}(x) = \frac{x + 5}{3}.$

Step 3: Finding $(f^{-1} \circ g^{-1})(x)$

This composition means applying $g^{-1}(x)$ first and then applying $f^{-1}(x)$ to the result. So: $(f^{-1} \circ g^{-1})(x) = f^{-1}\left(g^{-1}(x)\right) = f^{-1}\left(\frac{x + 5}{3}\right).$ Now substitute $f^{-1}(x) = x - 3$: $f^{-1}\left(\frac{x + 5}{3}\right) = \frac{x + 5}{3} - 3.$ Simplifying: $\frac{x + 5}{3} - 3 = \frac{x + 5}{3} - \frac{9}{3} = \frac{x - 4}{3}.$ Thus: $(f^{-1} \circ g^{-1})(x) = \frac{x - 4}{3}.$

Step 4: Finding $(g^{-1} \circ f^{-1})(x)$

This composition means applying $f^{-1}(x)$ first and then applying $g^{-1}(x)$ to the result. So: $(g^{-1} \circ f^{-1})(x) = g^{-1}(f^{-1}(x)) = g^{-1}(x - 3).$ Now substitute $g^{-1}(x) = \frac{x + 5}{3}$: $g^{-1}(x - 3) = \frac{(x - 3) + 5}{3} = \frac{x + 2}{3}.$ Thus: $(g^{-1} \circ f^{-1})(x) = \frac{x + 2}{3}.$

Final Answers:

• $f^{-1}(x) = x - 3$
• $g^{-1}(x) = \frac{x + 5}{3}$
• $(f^{-1} \circ g^{-1})(x) = \frac{x - 4}{3}$
• $(g^{-1} \circ f^{-1})(x) = \frac{x + 2}{3}$

Would you like further details or have any questions?

Here are five related questions you can explore:

1. How do we find the inverse of more complex functions?
2. What is the graphical interpretation of inverse functions?
3. Can every function have an inverse?
4. How do you determine if a function is one-to-one and invertible?
5. How do compositions of inverse functions behave in general?

Tip: Always ensure the function is one-to-one before finding its inverse, as only one-to-one functions have inverses.