Let f and g be the functions from the set of integers to the set integers…
2013
Let f and g be the functions from the set of integers to the set integers defined by
\(f(x) = 2x + 3 \text{ and} \ g(x) = 3x + 2 \)
Then the composition of f and g and g and f is given as
- A.
\(6x + 7, 6x + 11 \) - B.
\(6x + 11, 6x + 7 \) - C.
\(5x + 5, 5x + 5\) - D.
None of the above
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Correct answer: A
Compute the compositions by substitution.
Compute f(g(x)) (apply g, then f):
g(x) = 3x + 2, so f(g(x)) = 2*(3x + 2) + 3 = 6x + 4 + 3 = 6x + 7.
Compute g(f(x)) (apply f, then g):
f(x) = 2x + 3, so g(f(x)) = 3*(2x + 3) + 2 = 6x + 9 + 2 = 6x + 11.
Therefore the compositions are f∘g = 6x + 7 and g∘f = 6x + 11.
Note: composition is not commutative in general, so f∘g and g∘f can be different.
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