5.1.1 Combining Functions - Part I
Two functions:
x
1
2
f(x)
2
6
g(x)
3
0
3
4
-1
New function, which we call f + g:
x
1
2
3
(f+g)(x) 5
6
3
 inputs (domain) f + g:
 outputs:
Definition:
same as f and g
add outputs of f and g
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f . g)(x) = f(x) . g(x)
(f / g)(x) = f(x) / g(x)
 if either f or g is undefined for some number, any combined
function must be undefined for that number
 if g(a) = 0 for some value a, then (f /g)(a) is undefined
(why?).
Example: let f(x) = 2x + 3 and g(x) = x2 - 4
(f + g)(x) = f(x) + g(x) = (2x + 3) + (x2 - 4) = x2 + 2x - 1
5.1.1-1
Composition of Functions
A new operation on functions, with a new operation symbol: o
Notation:
Pronunciation:
Definition:
fog
"f composed with g"
(f o g)(x) = f(g(x))
“f of g of x”
Example:
f(x) = 2x + 1, g(x) = x2
(f o g)(x) = f(g(x)) = f(x2) = 2x2 + 1
so:
(f o g)(x) = 2x2 + 1
(Note that this is no more than brute-force substitution!!)
and: (f o g)(3) = 2(32) + 1 = 19
________________________________________________
Getting there quicker
Suppose:
(1) you don't need the general formula for a composition
i. e.
(f o g)(x) = 2x2 + 1
(2) you just want to compute one output: e.g. (f o g)(3)
Do it the easy way!
Don’t bother to compute the general formula first!
Just do:
(f o g)(3) = f(g(3)) = f(9) = 19
5.1.1-2