A Quick Review of Composite Functions and Inverse Functions

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.