Question:
If \( f(x) = 3x + 6 \), \( g(x) = 4x + k \), and \( f \circ g(x) = g \circ f(x) \), then find \( k \):
If \( f(x) = 3x + 6 \), \( g(x) = 4x + k \), and \( f \circ g(x) = g \circ f(x) \), then find \( k \):
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When solving problems with function compositions, equate the two expressions and solve for the unknowns. Pay attention to the structure of the functions to avoid mistakes.
Updated On: Apr 15, 2025
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The Correct Option is C
Solution and Explanation
Given the functions \( f(x) = 3x + 6 \) and \( g(x) = 4x + k \) and that \( f \circ g(x) = g \circ f(x) \), meaning:
\[
f(g(x)) = g(f(x)).
\]
First compute both compositions:
\[
f(g(x)) = f(4x + k) = 3(4x + k) + 6 = 12x + 3k + 6.
\]
\[
g(f(x)) = g(3x + 6) = 4(3x + 6) + k = 12x + 24 + k.
\]
Then, equate the two expressions:
\[
12x + 3k + 6 = 12x + 24 + k.
\]
Canceling out the \( 12x \) terms:
\[
3k + 6 = 24 + k.
\]
Solve for \( k \):
\[
3k - k = 24 - 6,
\]
\[
2k = 18,
\]
\[
k = 9.
\]
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