### 1.7 Combination of Functions

```1.7 Combination of Functions
Add, Subtract, Multiply and Divide functions
Understanding combination functions
Sum and Difference
Let f(x) = 2x + 5
and g(x) = x2-3
Sum f(x) + g(x) = 2x + 5 + x2-3
thus (f + g)(x) = x2 + 2x + 2
Difference
(f - g)(x) = 2x + 5 – (x2 – 3)
= -x 2 + 2x + 8
Product and Quotient
Let f(x) = 2x + 5
Product
Quotient
and g(x) = x2-3
f(x)g(x) so (fg)(x) = (2x + 5)(x2 – 3)
= 2x3 + 5x2 – 6x – 15
g(x)≠0
So what is the domain of the Quotient
function?
So what is the domain of the Quotient
function?
All reals that does not make the denominator zero.
What about a square root?
Let
 The Domain of
 The Domain of
and
is [0, ∞)
[ - 5, 5 ]
Domain is [0, 5) ; Why?
 The zero comes from
can not have negative
numbers,
 but
must have numbers less then 5.
Since it is in the denominator g(x) can not be zero.
How would the Domain change if
The Domain would be (0, 5]
What cause the difference?
Composition of Function
Composition of function is where the range of one function
become the domain of the other function.
Let f(x) = x3 + 1
and g(x)= x + 4
Old way of written a composition was f(g(x))
New way
(f∘g)(x) = (x+4)3 + 1
(g∘f)(x) = (x3+1)+4
(f∘g)(x) = (x+4)3 + 1
(x + 4)3
= (x + 4)(x + 4)(x + 4)
=(x2 + 8x + 16)(x + 4)
=(x3 + 8x2 + 16x)+(4x2 + 32x + 64)
= x3 + 12x2 + 48x + 64
(f∘g)(x) = x3 + 12x2 + 48x + 64 + 1
= x3 + 12x2 + 48x + 65
Let f(x)= x2 + 8 and
Domain of f(x) is ( - ∞,∞)
Domain of g(x) is [- 4, 4]
So the Domain would domain in which both
equations work. [ -4, 4]
Why use Composition functions?
To break functions into smaller easier to handle parts.
h(x)= (f∘g)(x)
Into 2 equations f(x) = 1/x and g(x) =(x+ 3)2
or f(x) = (x + 3)-2 and g(x) = (x + 3)
Homework
 Page 74 – 76
# 1, 9, 19, 27, 33, 39, 43, 47, 51, 56, 60 ,67
Homework
 Page 74 – 76
# 5, 13, 22, 31, 35, 41, 45, 49, 53, 57,63
```