Let's go through each part of the problem based on the image.

We have two functions $f(x)$ and $g(x)$ given in tabular form:

• $f(x)$:

  • $f(1) = 4$
  • $f(2) = 1$
  • $f(3) = 5$
  • $f(4) = 5$
  • $f(5) = 3$

• $g(x)$:

  • $g(1) = 4$
  • $g(2) = 4$
  • $g(3) = 2$
  • $g(4) = 2$
  • $g(5) = 3$

(a) What is $f(3)$?

From the table for $f(x)$, we see that $f(3) = 5$.

(b) Compute $(f \circ g)(5)$, i.e., $f(g(5))$.

• First, find $g(5)$: From the table for $g(x)$, we know that $g(5) = 3$.
• Now, substitute into $f(x)$: $f(g(5)) = f(3)$.
• From part (a), $f(3) = 5$.

Thus, $(f \circ g)(5) = 5$.

(c) Compute $(g \circ f)(3)$, i.e., $g(f(3))$.

• First, find $f(3)$: From part (a), $f(3) = 5$.
• Now, substitute into $g(x)$: $g(f(3)) = g(5)$.
• From the table for $g(x)$, $g(5) = 3$.

Thus, $(g \circ f)(3) = 3$.

(d) Compute $(f^{-1} \circ g)(2)$, i.e., $f^{-1}(g(2))$.

• First, find $g(2)$: From the table for $g(x)$, $g(2) = 4$.
• Now, find $f^{-1}(4)$. This means we are looking for the value of $x$ such that $f(x) = 4$. From the table for $f(x)$, $f(1) = 4$, so $f^{-1}(4) = 1$.

Thus, $(f^{-1} \circ g)(2) = 1$.

Would you like more details or clarifications on any of the steps?

Here are 5 related questions for practice:

1. What is $g(f(4))$?
2. Compute $(f \circ g)(3)$.
3. Find $f^{-1}(5)$.
4. What is $(g \circ g)(1)$?
5. Compute $f(g(4))$.

Tip: For composite functions like $(f \circ g)(x)$, always evaluate the inside function first, then apply the outside function!