If f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)} then gof = {(1, 3), (3, 1), (4, 3)}Group startsTrue or FalseTrue, unselectedFalse

To determine whether $g \circ f = \{(1, 3), (3, 1), (4, 3)\}$ is true or false, we need to find the composite function $g \circ f$ by applying the function $f$ first, and then applying $g$ to the results.

Step 1: Break Down the Problem

• We will use the pairs in $f$ to find the resulting outputs.
• Then, for each output from $f$, we will use those as inputs for $g$.

Step 2: Relevant Concepts

• The notation $g \circ f$ means applying $f$ first and then applying $g$ to the results of $f$.

Step 3: Analysis and Detail

1. Apply $f$:

   • From $f = \{(1, 2), (3, 5), (4, 1)\}$:
     • $f(1) = 2$
     • $f(3) = 5$
     • $f(4) = 1$

2. Apply $g$:

   • Now, we take the outputs from $f$ and apply $g$:
   • For $f(1) = 2$:
     • $g(2) = 3$ → gives the pair $(1, 3)$
   • For $f(3) = 5$:
     • $g(5) = 1$ → gives the pair $(3, 1)$
   • For $f(4) = 1$:
     • $g(1) = 3$ → gives the pair $(4, 3)$

Combining these results, we find: $g \circ f = \{(1, 3), (3, 1), (4, 3)\}$

Step 4: Verify and Summarize

• We have matched each input from $f$ and found the corresponding outputs via $g$.
• The resulting composite function $g \circ f$ indeed equals $\{(1, 3), (3, 1), (4, 3)\}$.

Final Answer

True