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 \(5a+5b\).
+128475 g(x) = 5x - 4
g(x) = f-1(x) - 3
g(x) + 3 = f-1(x)
If f(x) = ax + b
Then
y =ax + b
y - b = ax
(y - b) / a = x
(x - b) / a = y = f-1(x)
So
g(x) + 3 = (x - b) / a
5x - 4 + 3 = (1/a)x - b/a
5x - 1 = (1/a)x - b/a
So
5 = 1/a
a =1/5
And
-1 = -b/a
-1 = -b / (1/5)
1 = b / (1/5)
b =1/5
So
5a + 5b =
5(1/5) + 5(1/5) =
1 + 1 =
2
