noong unang panahon may mga senior high schools na masipag mag aral
sila ay sina Arvie at Jessica. Pumunta sila sa library para maka libre ng internet kasi poor sila. haha. :D
ito ang kanilang nakalap (Lol. XD):
Operations on Functions
Functions with overlapping domains can be added, subtracted, multiplied and divided. If f(x)f(x) and g(x)g(x) are two functions, then for all xx in the domain of both functions the sum, difference, product and quotient are defined as follows.
(f+g)(x)=f(x)+g(x)(f−g)(x)=f(x)−g(x)(fg)(x)=f(x)×g(x)(fg)(x)=f(x)g(x),g(x)≠0
Example :
Let f(x)=2x+1f(x)=2x+1 and g(x)=x2−4g(x)=x2−4
Find (f+g)(x),(f−g)(x),(fg)(x)(f+g)(x),(f−g)(x),(fg)(x) and (fg)(x)(fg)(x).
(f+g)(x)=f(x)+g(x) =(2x+1)+(x2−4) =x2+2x−3(f+g)(x)=f(x)+g(x) =(2x+1)+(x2−4) =x2+2x−3
(f−g)(x)=f(x)−g(x) =(2x+1)−(x2−4) =−x2+2x+5(f−g)(x)=f(x)−g(x) =(2x+1)−(x2−4) =−x2+2x+5
(fg)(x)=f(x)×g(x) =(2x+1)(x2−4) =2x3+x2−8x−4(fg)(x)=f(x)×g(x) =(2x+1)(x2−4) =2x3+x2−8x−4
(fg)(x)=f(x)g(x)=2x+1x2−4,x≠±2
Another way to combine two functions to create a new function is called the composition of functions. In the composition of functions we substitute an entire function into another function.
The notation of the functionff with gg is (f∘g)(x)=f(g(x))(f∘g)(x)=f(g(x))and is read ff of gg of xx. It means that wherever there is an xx in the function ff, it is replaced with the function g(x)g(x). The domain of f∘gf∘g is the set of all xx in the domain of gg such that g(x)g(x) is in the domain of ff.
Example 1:
Let f(x)=x2f(x)=x2 and g(x)=x−3g(x)=x−3. Find f(g(x))f(g(x)).
f(g(x))=f(x−3) =(x−3)2 =x2−6x+9
Example 2:
Let f(x)=2x−1f(x)=2x−1 and g(x)=x+2g(x)=x+2. Find f(g(x))f(g(x)).
f(g(x))=f(x+2) =2(x+2)−1 =2x+3
Order DOES matter when finding the composition of functions.
Example 3:
Let f(x)=3x+1f(x)=3x+1 and g(x)=2x−3g(x)=2x−3.
Find f(g(x))f(g(x)) and g(f(x))g(f(x)).
f(g(x))=f(2x−3) =3(2x−3)+1 =6x−8g(f(x))=f(3x+1) =2(3x+1)−3 =6x−1f(g(x))=f(2x−3) =3(2x−3)+1 =6x−8g(f(x))=f(3x+1) =2(3x+1)−3 =6x−1
Since 6x−8≠2x−1, f(g(x))≠g(f(x))6x−8≠2x−1, f(g(x))≠g(f(x)).
Source:
http://hotmath.com/hotmath_help/topics/operations-on-functions.html
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hahahahaha. ugaling copy paste. >:D
Ganyan ang masipag! :P
