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 $g \circ f$.

Step 1: Understand Composite Function

The notation $g \circ f$ represents the composite function of $g$ and $f$. This means we first apply the function $f$ to an input $x$, and then apply the function $g$ to the result of $f(x)$. The domain of $g \circ f$ is the domain of $f$, which is $\{2{,}3{,}4{,}5\}$.

Step 2: Calculate for $x = 2$

First, we find the value of $f(2)$. From the given definition, $f(2)=3$. Then, we substitute this value into $g$, so we find $g(3)$. From the given definition, $g(3)=7$. Therefore, $(g \circ f)(2) = 7$.

Step 3: Calculate for $x = 3$

Next, we find the value of $f(3)$. From the given definition, $f(3)=4$. Then, we substitute this value into $g$, so we find $g(4)$. From the given definition, $g(4)=7$. Therefore, $(g \circ f)(3) = 7$.

Step 4: Calculate for $x = 4$

Now, we find the value of $f(4)$. From the given definition, $f(4)=5$. Then, we substitute this value into $g$, so we find $g(5)$. From the given definition, $g(5)=11$. Therefore, $(g \circ f)(4) = 11$.

Step 5: Calculate for $x = 5$

Finally, we find the value of $f(5)$. From the given definition, $f(5)=5$. Then, we substitute this value into $g$, so we find $g(5)$. From the given definition, $g(5)=11$. Therefore, $(g \circ f)(5) = 11$.

Step 6: Combine Results

By combining all the calculated values, we can define the composite function $g \circ f$ as a set of ordered pairs, where each pair is $(x, (g \circ f)(x))$.

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