1. Precalc 3.5 - Combinations of Functions
ex: f(x) = (2x - 3) g(x) = (x2 - 1)
(sum) : f(x) + g(x) = (2x - 3) + (x2 - 1) = x2 + 2x - 4
(diff) : f(x) - g(x) = (2x - 3) - (x2 - 1) = -x2 + 2x - 2
(prod) : f(x)•g(x) = (2x - 3)(x2 - 1) = 2x3 - 3x2 - 2x + 3
(quo) : f(x)
g(x)
(2x - 3)
(x2 - 1) =
(x  + 1)

2. ex:
f(x) = (2x + 1) g(x) = (x2 + 2x - 1)
Find (f + g)(x)
Evaluate sum at x = 2
(f + g)(x) = f(x) + g(x) = 2x x2 + 2x - 1 = x2 + 4x
(f + g)(2) = (2)2 + 4(2) = = 12
Find (f - g)(x)
Evaluate difference at x = 2
(f - g)(x) = f(x) - g(x) = 2x (x2 + 2x - 1) =
2x x2 - 2x + 1
= -x2 + 2
(f - g)(2) = -(2)2 + 2 = = -2

3. Mixed Domains
ex: Find the Domain of (f + g): f g
(f + g)
f(x) = 1 x
g(x) = x
x > 0
x  0
x > 0

4. ex: Find the Domain of f/g and g/f:f g 2 -2
f/g f g 2 -2
f/g
f(x) = x
g(x) = x2
x > 0
f(x)
g(x) = x
4 - x2
0< x < 2
0 < x < 2
-2 < x < 2 = x
4 - x2
f(x)
g(x)
-2 < x < 2
x > 0

5. (f g) 3.6 continued - Compositions ex: f(x) = x2 and g(x) = x + 1
Notation: (f o g)(x) - say: “f-circle-g of x”
f(x) = x2
(f g)
f [g(x)] = [g(x)]2
f(x + 1)
= (x + 1)2
Take function g…
(x + 1)
= x2 + 2x + 1
And put it into function f…

6. ex: Find (f o g)(x) for f(x) = x and g(x) = (x - 1)
(f o g)(x) = f [g(x)] = g(x)
= x - 1
Domain: x > 1
[1, )
ex: Find (f o g)(2) =
= 1 = 1
ex: Find (f o g)(0)
0 Not in Domain (x > 1)

7. ex: For f(x) = (x + 2) and g(x) = (4 - x2)
- Find (f o g)(x) and (g o f)(x)
(f o g)(x) = f [g(x)]
= f(4 - x2)
= (4 - x2 + 2)
= (- x2 + 6)
(g o f)(x) = g [f(x)]
= g(x + 2)
= [4 – (x + 2) 2]
= 4 – (x2 + 4x + 4)
= 4 - x2 - 4x - 4
= - x2 - 4x

8. Domains of Compositions1 x
x - 3
ex:
f (x) =
g (x) =
x + 2
What is the domain of (f o g)(x) ? x
x - 3
= -2 -2 1 =
x - 3 1
+ 2 x
(f o g)(x) = f [ g(x) ] =
(solve)
-2(x - 3) = x
-2x + 6 = x
x ≠ 3 x
x - 3
≠ -2
x ≠ 3
6 = 3x
x ≠ 2
x = 2
Domain of (f o g)(x) = (x ≠ 2, 3)

9. Domains of Compositions (Summary / Quick Tips)
f [ g(x) ]
1) Any restrictions on the inside function will stay
2 is BAD for f!
2) Anything bad for f that the inside function might produce has to be solved for and excluded…
g (x) = x
f (x) =
x - 2 1
ex:
x ≠ 2
g (x) ≠ 2
9 - x ≠ 2
x ≠ 5
x < 9
So, for (f o g)(x) …
x < 9, ≠ 5

10. (3x – 5)3 Decomposing Composites
ex: Break h(x) = (3x – 5)3 into 2 functions f and g such that
(f o g)(x) = h(x)
(3x – 5)3
f (x) = x3
outside function
inside function
g(x) = 3x - 5
= f(x)
(f o g)(x) = (3x – 5)3
= g(x)
= h(x)

11. 1 (x – 2) 2 Decomposing Composites
ex: Break h(x) = into 2 functions f and g such that
(f o g)(x) = h(x)
(x – 2)2 1 1
(x – 2) 2 1
f (x) = x2
outside function
g(x) = x - 2
inside function 1
= f(x)
(f o g)(x) =
(x – 2)2
= g(x)
= h(x)