Let $f: \{2, 3, 4, 5\} \rightarrow \{3, 4, 5, 9\}$ and $g: \{3, 4, 5, 9\} \rightarrow \{7, 11, 15\}$ be functions defined as $f(2)=3, f(3)=4, f(4)=f(5)=5$ and $g(3)=g(4)=7$ and $g(5)=g(9)=11$. Find $g \circ f$.
The composition $g \circ f$ is defined as $(g \circ f)(x) = g(f(x))$.
We calculate the values for each element in the domain $\{2, 3, 4, 5\}$:
$(g \circ f)(2) = g(f(2)) = g(3) = 7$
$(g \circ f)(3) = g(f(3)) = g(4) = 7$
$(g \circ f)(4) = g(f(4)) = g(5) = 11$
$(g \circ f)(5) = g(f(5)) = g(5) = 11$
Thus,$g \circ f = \{(2, 7), (3, 7), (4, 11), (5, 11)\}$.
We calculate the values for each element in the domain $\{2, 3, 4, 5\}$:
$(g \circ f)(2) = g(f(2)) = g(3) = 7$
$(g \circ f)(3) = g(f(3)) = g(4) = 7$
$(g \circ f)(4) = g(f(4)) = g(5) = 11$
$(g \circ f)(5) = g(f(5)) = g(5) = 11$
Thus,$g \circ f = \{(2, 7), (3, 7), (4, 11), (5, 11)\}$.
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