COMPOSITION OF FUNCTIONS & INVERSES

Composition of Functions

Another way of denoting and performing SUBSTITUTION

The notatation for: f(x)

This DOES NOT MEAN MULITPLICATION.

Example 1

Given f(x) = 3x - 2, find f(2).

Steps

Substitute 2 for x

f(2) = 3(2) - 2 = 6 - 2 = 4

Example 2

Given g(x) = x^2 - x, find g(-3)

g(-3) = (-3)^2 - (-3) = 9 +3 =    9 + 3 = 12

g(-3) = 12

Example 3

Given g(x) = 3x - 4x2 + 2, find g(5)

g(5) = 3(5) - 4(5)2 + 2 =

15 - 4(25) + 2 = 15 - 100 + 2

= -83

g(5) = -83

Given f(x) = x - 5, find f(a+1)

f(a + 1) = (a + 1) - 5 = a+1 - 5

]f(a + 1) = a – 4

How do you compose two functions?

Just the same we will still be replacing x with whatever we have in the parentheses.

The notation looks like g(f(x)) or f(g(x)). 

We read it ‘g of f of x’ or ‘f of g of x’

 EXAMPLE 4

Given f(x) = 2x + 2 and     g(x) = 2, find f(g(x)).

Start on the inside. f(g(x))

g(x) = 2, so replace it. 

f(g(x)) = f(2) = 2(2) + 2 = 6

INVERSE OF A FUNCTION

In mathematics, an inverse function is a function that undoes another function: If the function f applied to an input x gives a result of y, then applying the inverse function g to y gives the resultx, and vice versa. i.e. f(x) = y, and g(y) = x. More directly, g(f(x)) = x, meaning g composed with f form an identity.

A function f that has an inverse is called invertible; the inverse function is then uniquely determined by f and is denoted by f −1, read f inverse. Superscripted "−1" does not refer to numericalexponentiation: see composition monoid for explanation of this notation

-source- wikipedia

DIAGRAMS OF INVERSES



FINDING INVERSES
Steps:
1. Rewrite using x and y
2. Switch x and y
3. Solve for y
4. The y is your inverse. Express it as f^-1(x)

Example:

Find the inverse of f(x)=3x+1

write it as             y=3x+1
switch x and y      x=3y+1
Solve for y         3y=x-1
                           y=(x-1)/3
Rename               f^-1=(x-1)/3



VIDEO TUTORIALS ON COMPOSITION OF FUNCTIONS

2. Composition of Functions  pt I

3. Composition of Function pt II 

4. Inverses and Composition pt I

5. Inverses and Composition pt II 

6. Evaluating and Composition of Functions

7. Verifying Inverses by Composition

8. Finding the Domain and Range of Radical Functions
Mathgotserved
\

VERIFYING INVERSES BY COMPOSITION WORKED OUT EXAMPLES