Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, find $5a+5b$.
+9676 First, we find \(f^{-1}(x)\) in terms of a and b.
\(f(x) = ax + b\\ x = af^{-1}(x) + b\\ f^{-1}(x) = \dfrac{x - b}a\)
Now,
\(g(x) = f^{-1}(x) - 3\\ 5x - 4 = \dfrac xa - \left(\dfrac ba + 3\right)\)
Comparing coefficients:
\(\dfrac1a = 5\\ a = \dfrac15\)
\(5b + 3 = 4\\ b = \dfrac15\)
Therefore
\(5a + 5b = 5\cdot \dfrac15 + 5\cdot \dfrac15 = 2\)
