Composite Functions
Given f(x) and g(x), we have:
Adding: (f+g)(x) = f(x) + g(x)
Subtracting: (f-g)(x) = f(x) - g(x)
Multiplying: (f.g)(x) = f(x) . g(x)
Dividing: (f/g)(x) = [f(x)]/[g(x)]
Substitution: (f˚g)(x) = f(x)˚g(x) = f(g(x))
Example:
f(x) = x^2 + 1 and g(x) = x-1
(f+g)(x) = x^2 + 1 + x - 1 = x^2 + x
(f-g)(x) = x^2 + 1 - (x-1) = x^2 - x + 2
(f.g)(x) = (x^2+1)(x-1)= x^3 - x^2 + x - 1
(f/g)(x) = (x^2+1)/(x-1)
(f˚g)(x) = f(g(x)) = f(x-1) = (x-1)^2 + 1
Inverse Functions
Two functions are defined as inverses if the compositions f(g(x))=x and g(f(x))=x are true.
Two ways to find inverse functions:
1. Graph the function, then reflect it into y=x to get the graph of the inverse function.
2. Switch the x's for y's and the y's for x's in the function, solve for y to get the inverse function.
