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.
I don't know how to do this.
+23246 First: given f(x) = ax + b, find f-1(x):
1) let y = f(x) ---> y = ax + b
2) interchange x and y ---> x = ay + b
3) solve for y: ---> x - b = ay ---> y = (x - b) / a
4) exchange f-1(x) for y ---> f-1(x) = (x - b)/a
Since g(x) = f-1(x) - 3 ---> g(x) = (x - b)/a - 3
Since g(x) = 5x - 4 ---> 5x - 4 = (x - b)/a - 3
Rewriting: 5x - 4 = x/a - b/a - 3
Setting the x-terms equal to each other: 5x = x/a
---> 5ax = x
---> 5a = 1
---> a = 1/5
Setting the numbers equal to each other: -4 = -b/a - 3
---> -1 = -b/a
---> -1a = -b
---> a = b
---> b = 1/5
