Let's go through each part of the question using the given tables for the functions $f$ and $g$:

Tables for Functions $f$ and $g$:

• Function $f$:
  • $f(1) = 2$
  • $f(2) = -3$
  • $f(3) = -1$
  • $f(4) = 3$
• Function $g$:
  • $g(1) = 5$
  • $g(2) = 5$
  • $g(3) = 3$
  • $g(4) = -1$

Solution:

(a) Write down the values of $g(3)$ and $f^{-1}(3)$:

• $g(3)$: From the table for $g$, $g(3) = 3$.
• $f^{-1}(3)$: To find $f^{-1}(3)$, we need to find $x$ such that $f(x) = 3$. From the table for $f$:
  • $f(4) = 3$ So, $f^{-1}(3) = 4$.

(b) Calculate $(f \circ g)(2)$:

• $(f \circ g)(2) = f(g(2))$
• From the table, $g(2) = 5$.
• We need $f(5)$, but the table for $f$ only gives values for $x = 1, 2, 3, 4$. So, we can't calculate $f(5)$ since it's not given in the table.

(c) Calculate $(g \circ g)(3)$:

• $(g \circ g)(3) = g(g(3))$
• From the table, $g(3) = 3$.
• Now, $g(3) = 3$ again, so: $g(g(3)) = g(3) = 3$.

(d) Find a solution of the equation $(g \circ f)(x) = 3$:

• $(g \circ f)(x) = g(f(x))$

• We need to find $x$ such that $g(f(x)) = 3$.

  • $f(1) = 2$ → $g(f(1)) = g(2) = 5$ (Not 3)
  • $f(2) = -3$ → Not in the table for $g$
  • $f(3) = -1$ → Not in the table for $g$
  • $f(4) = 3$ → $g(f(4)) = g(3) = 3$

  So, $x = 4$ is a solution.

Summary of Solutions:

• (a) $g(3) = 3$, $f^{-1}(3) = 4$
• (b) $(f \circ g)(2)$: Cannot be determined from the given information.
• (c) $(g \circ g)(3) = 3$
• (d) $x = 4$

Would you like further details or have any questions about these calculations?

Here are some related questions to explore:

1. How do you find the inverse of a function using its table of values?
2. What does the composition of functions represent?
3. How would you determine $(f \circ g)(x)$ if the tables provided values for all integers?
4. What if $g$ had a different range? How would it affect finding $(g \circ f)(x) = 3$?
5. How can you use function composition in real-life problem solving?

Tip: When working with tables of values for functions, always ensure that the function's domain and range fit within the given data to avoid undefined situations.