Articles

Compound function


Let's look at an example to understand what a composite function is. Consider the sets:

A = {- 2, -1,0,1,2}
B = {- 2,1,4,7,10}
C = {3,0,15,48,99}

And the functions:
f: AB defined by f (x) = 3x + 4
g: BC defined by g (y) = y2-1

As shown in the diagram above for all x A we have a single y B such that y = 3x + 4, and for all y B there is a single z C such that z = y2-1. Then we conclude that there is a function h of A in C, defined by h (x) = z or h (x) = 9x2+ 24x + 15 because:
h (x) = z h (x) = y2-1
And where y = 3x + 4, then h (x) = (3x + 4)2-1 h (x) = 9x2+ 24x + 15.

The function h (x) is called function composed of g with f. We can indicate it by g o f (we read “g composed with f”) or gf (x) (we read “g of f of x”). Let's look at some exercises to better understand the idea of ​​compound function.

Exercises solved

1) Given the functions f (x) = x2-1 and g (x) = 2x, calculate fg (x) and gf (x).
Resolution:
fg (x) = f (2x) = (2x)2-1 = 4x2-1
gf (x) = g (x2-1) = 2 (x2-1) = 2x2-2

2) Given the functions f (x) = 5x and fg (x) = 3x + 2, calculate g (x).
Resolution:
Since f (x) = 5x, then fg (x) = 5.g (x).
However, fg (x) = 3x + 2, therefore:
5.g (x) = 3x + 2, hence g (x) = (3x + 2) / 5

3) Given the functions f (x) = x2+1 and g (x) = 3x-4, determine fg (3).
Resolution: g (3) = 3.3-4 = 5 fg (3) = f (5) = 52+1 = 25+1= 26.

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