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))

compositionoffunq1.png

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.

compositionoffunq1p1.png

The value of g(2) is 0.

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

compositionoffunq1p1.png

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

evaluatingcompofunfromtableq1

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

compositionfromarrowdiaramq1

Find  f∘f  

i) f∘f :

Solution :

Input is a :

f∘f (a) = f [f(a)]

= f[b]

= c

So, (a, c)

Input is b :

f∘f (b) = f [f(b)]

= f[c]

= b

So, (b, b)

Input is c :

f∘f (c) = f [f(c)]

= f[b]

= c

So, (c, c)

Input is d :

f∘f (d) = f [f(d)]

= f[d]

= d

So, (d, d)

compositionoffunctionq1

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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