by: dr. julia arnold composition is a binary operation like addition, subtraction, multiplication...
TRANSCRIPT
By: Dr. Julia Arnold
Composition is a binary operation like addition , subtraction, multiplication and division are binary operations. (meaning they operate on two elements)
f+g
f-g
fg g
f
The composition symbol is: Thus gf
The easiest way to describe composition is to say it is like substitution. In fact
))(()( xgfxgf
Read f of g of x which means substitute g(x) for x in the f(x) expression.
For example:Suppose f(x)= 2x + 3, and
g(x) = 8 - xThen ))(()( xgfxgf
Means substitute the g function forx in the f function… like this
f(x)= 2x + 3
f(g(x) )= 2 g(x) + 3
f(x)= 2x + 3, and g(x) = 8 - x ))(()( xgfxgf
f(x)= 2x + 3
f(g(x) )= 2 g(x) + 3Nowsubstitute what g equals for g(x)
f(8 - x)= 2 (8 - x) + 3
= 16 - 2x + 3= 19 - 2x
So, = 19 - 2x xgf
An interesting fact is that
xfgxgf most of the time.
Let’s see if this is thecase for theprevious example.
f(x) = 2x + 3, andg(x) = 8 - x
))(( xfgxfg Thus we will substitute f into g.
g(x) = 8 - x
g(f(x) ) = 8 - f(x)
Nowsubstitutewhat f(x) is:
g(2x + 3) = 8 - (2x + 3)
= 8 - 2x - 3= 5 - 2x
Let 32)( 2 xxxfand
xxg )(
32)( 2 xxxf xxg )(
))(()( xgfxgf
Write the f function 32)( 2 xxxfSubstitute g(x) for x 3)(2)())(( 2 xgxgxgf
Replace g(x) with x 32)(2
xxxf
Simplify 32))(( xxxgf
Step 1
Step 2
Step 3
Step 4
32)( 2 xxxf xxg )(
Find: xfg When ready click your mouse.
The answer is:
A) 322 xx
B) 32 xxMove your mouse overthe correct answer.
xxxf 2)(xx
xg
2
1)(
Find: xgf When ready click your mouse.
The answer is:
B) xxxx
2
2
2
11
A)xxxx
224
11
Move your mouse overthe correct answer.
1)( xxf 21
)( x
xg
Find: xfg When ready click your mouse.
The answer is:
B) 2
1
1
x
A) 121
121
x
x
xMove your mouse overthe correct answer.
1)( xxf 21
)( x
xg
A) 121
121
x
x
x
Ans. A for the previous example
Was actually xgf
Do the practice problems followingThis module in your module section Of Blackboard. Check your work usingThis live math page.