Question

If \( f(x) = 3x + 6 \), \( g(x) = 4x + k \), and \( f \circ g(x) = g \circ f(x) \), then find \( k \):

• A. 3
• B. 6
• C. 9
• D. 12

Hint:

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.

Answer 1

The Correct Option is C

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 \( k \): \[ 3k - k = 24 - 6, \] \[ 2k = 18, \] \[ k = 9. \]