Let f:{2,3,4,5}→{3,4,5,9} and g:{3,4,5,9}→{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∘f.
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Step-by-Step Solution
Step 1: Understand Composite Function
The notation g∘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∘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∘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∘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∘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∘f)(5)=11.
Step 6: Combine Results
By combining all the calculated values, we can define the composite function g∘f as a set of ordered pairs, where each pair is (x,(g∘f)(x)).