\(Let $f(x) = ax+b$, where $a$ and $b$ are real constants, and $g(x) = 2x - 5$. Suppose that for all $x$, it is true that $g(f(x)) = 3x + 4$. What is $a+b$?\)
f(x) = ax +b
g(x) = 2x - 5
g(f(x)) = 3x + 4
So....we have that
2 ( ax + b) - 5 = 3x + 4
2ax + 2b - 5 = 3x + 4
So....equating coefficients
2a = 3 and 2b - 5 = 4
a = 3/2 2b = 9
b = 9/2
So
a + b = (3/2) + (9/2) = 12 / 2 = 6
