### Instruction: 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 $a+b$?### Response: This problem involves two linear functions and their composition, which is another linear function.
I want to find a way to relate $f(x)$ and $g(x)$ to $g(f(x))$, and use that to find $a$ and $b$.
One way to do that is to plug in $f(x)$ into $g(x)$ and simplify.
So, $g(f(x)) = 2(ax+b) - 5 = 2ax + 2b - 5$.
This expression should be equal to $3x + 4$ for all $x$, so I can equate the coefficients of $x$ and the constant terms.
That means $2a = 3$ and $2b - 5 = 4$.
Solving for $a$ and $b$, I get $a = \frac{3}{2}$ and $b = \frac{9}{2}$.