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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 $5a+5b$.

 Jul 9, 2020
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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

\(5a + 5b = 5\cdot \dfrac15 + 5\cdot \dfrac15 = 2\)

 Jul 9, 2020

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