To find the value of \( c \) such that \( (f \circ g)(x) = (g \circ f)(x) \) for all \( x \), we first need to compute each composition separately.
Starting with \( f(x) = 3x - 4 \) and \( g(x) = 5x + c \):
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Calculate \( (f \circ g)(x) \): \[ (f \circ g)(x) = f(g(x)) = f(5x + c) \] Substitute \( 5x + c \) into \( f \): \[ f(5x + c) = 3(5x + c) - 4 = 15x + 3c - 4 \]
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Calculate \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(3x - 4) \] Substitute \( 3x - 4 \) into \( g \): \[ g(3x - 4) = 5(3x - 4) + c = 15x - 20 + c = 15x + (c - 20) \]
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Set \( (f \circ g)(x) = (g \circ f)(x) \): We now equate the two expressions: \[ 15x + 3c - 4 = 15x + (c - 20) \]
We can eliminate \( 15x \) from both sides: \[ 3c - 4 = c - 20 \]
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Solve for \( c \): Rearranging, we have: \[ 3c - c = -20 + 4 \] Simplifying gives: \[ 2c = -16 \] Dividing both sides by 2: \[ c = -8 \]
Thus, the value of \( c \) is \(\boxed{-8}\).