composite functions

Composite Function

"Function Composition" is applying one function to the results of another:

The result of f() is sent through g()It is written: (g º f)(x)

Which means: g(f(x))

Example: f(x) = 2x+3 and g(x) = x2

"x" is just a placeholder, and to avoid confusion let's just call it "input":f(input) = 2(input)+3

g(input) = (input)2

So, let's start:

(g º f)(x) = g(f(x))First we apply f, then apply g to that result:

(g º f)(x) = (2x+3)2

What if we reverse the order of f and g?

(f º g)(x) = f(g(x))First we apply g, then apply f to that result:

(f º g)(x) = 2x2+3

Operations on Functions

The sum f + g

xgxfxgf This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.

1432 32 xxgxxf

1432 32 xxgf

424 23 xx

Combine like terms & put in descending order

The difference f - g

xgxfxgf To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms.

1432 32 xxgxxf

1432 32 xxgf

1432 32 xx

Distribute negative

224 23 xx

The product f • g

xgxfxgf To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function.

1432 32 xxgxxf

1432 32 xxgf

31228 325 xxx

FOIL

Good idea to put in descending order but not required.

The quotient f /g

xgxf

xg

f

To find the quotient of two functions, put the first one over the second.

1432 32 xxgxxf

14

323

2

x

x

g

f Nothing more you could do here. (If you can reduce these you should).

COMPOSITION

OFFUNCTIONS

The Composition Function

xgfxgf This is read “f composition g” and means to copy the f function down but where ever you see an x, substitute in the g function.

1432 32 xxgxxf

314223 xgf

51632321632 3636 xxxx

FOIL first and then distribute the 2

xfgxfg This is read “g composition f” and means to copy the g function down but where ever you see an x, substitute in the f function.

1432 32 xxgxxf

132432 xfg

You could multiply this out but since it’s to the 3rd power we won’t

So the first 4 operations on functions are pretty straight forward.

The rules for the domain of functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g.

For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.

xffxff This is read “f composition f” and means to copy the f function down but where ever you see an x, substitute in the f function. (So sub the function into itself).

1432 32 xxgxxf

332222 xff

The DOMAIN of the Composition Function

The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

11

xxgx

xf

1

1

xgf

The domain of g is x 1

We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x 1 so the domain of the composition would be combining the two restrictions.

1: is ofdomain xxgf

0: yy 6: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

1

15

x

xgxxf

Hence we must exclude 6 from the domain of f(x)

For g(x) to cope with the output from f(x) we must ensure that the output does not include 1

5xxf

?)( xfg

1: yy

1

1

x

xg

1 x

0: yy 6: xx

The DOMAIN and RANGE of Composite Functions

Or we could find g o f (x) and determine the domain and range of the resulting expression.

1

15

x

xgxxf

However this approach must be used with CAUTION.

6

1)(

x

xfg

Domain: Range:

5: yy 1: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

1

15

x

xgxxf

Hence we must exclude 1 from the domain of g(x)

For f(x) to cope with the output from g(x) we must ensure that the output does not include 0

1

1

x

xg

?)( xgf

0: yy

5xxf

0 x

5: yy 1: xx

The DOMAIN and RANGE of Composite Functions

Or we could find f o g (x) and determine the domain and range of the resulting expression.

1

15

x

xgxxf

However this approach must be used with CAUTION.

51

1)(

x

xgf

Domain: Range:

0: yy

0: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

2xxgxxf

xxf

?)( xfg

0: yy

2xxg

0 x

0: yy 0: xx

The DOMAIN and RANGE of Composite Functions

Or we could find g o f (x) and determine the domain and range of the resulting expression.

2xxgxxf

However this approach must be used with CAUTION.

xxfg )(

Domain: Range:

Not: yandx

0: yy 2: xx

The DOMAIN and RANGE of Composite Functions

We could first look at the natural domain and range of f(x) and g(x).

22 xxgxxf

f o g (x) is a function for the natural domain of g(x)

f(x) can cope with all the numbers in the range of g(x) because the range of g(x) is contained within the domain of f(x)

2 xxg

?)( xgf

0: yy

xxf 2

0 x