# FIND COMPOSITION OF TWO FUNCTIONS

The composition of a function g with a function f is :

h(x) = g(f (x))

The domain of h is the set of all x-values such that x is in the domain of f and f (x) is in the domain of g.

Composition of function can be done in the following situations.

## Composition of Two Composition

Problem 1 :

f(x) = -9x + 3 and g(x) = x4, find (f g)(x)

Solution :

(f g)(x) = f(g(x))

The function g(x) is input of the function f(x).

= f(x4)

= -9(x4) + 3

(f g)(x) = -9x4 + 3

## Composition of Function From Graph

Problem 2 :

Use the graph of f(x) and g(x) to find the composition of functions.

f(g(2))

Solution:

f(g(2))

From the graphs given above, it is clear that the line at x = 2  is crossing the curve g(x) at the point 0.

The value of g(2) is 0.

f(g(2)) = f(0)

When x = 0, the vertical line crosses the curve f(x) at -3.

f(g(2)) = -3

## Composition from Table

Problem 3 :

Use the table of values to evaluate each expression

1)  f ( g(8) )     2)  f( g(5) )       3)  g ( f (5) )

Solution :

1)  f ( g(8) )

From the table, value of g(8) is 4.

f ( g(8) ) = f(4)

= 4

2)  f ( g(5) )

From the table, value of g(5) is 8.

f ( g(5) ) = f(8)

= 9

3)  g ( f (5) )

From the table, value of f(5) is 0.

g ( f(5) ) = f(0)

= 7

## Composition from Arrow Diagram

Problem 1 :

The arrow diagrams defines the functions

Find  f∘f

i) f∘f :

Solution :

 Input is a :f∘f (a) = f [f(a)]= f[b]= cSo, (a, c) Input is b :f∘f (b) = f [f(b)]= f[c]= bSo, (b, b) Input is c :f∘f (c) = f [f(c)]= f[b]= cSo, (c, c) Input is d :f∘f (d) = f [f(d)]= f[d]= dSo, (d, d)

## Composition of Three Functions

Given f(x) = -2x + 1, g(x) = x2, and h(x) = -1/2x + 1/2, evaluate the following:

Problem 4 :

Given f(x) = -2x + 1, find f(-6)

Solution:

f(x) = -2x + 1

f(-6) = -2(-6) + 1

= 12 + 1

f(-6) = 13

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