Composition Functions
Composition functions are functions that combine to make a new function. We use the no-
tation ◦ to denote a composition.
f ◦ g is the composition function that has f composed with g. Be aware though, f ◦ g is not
the same as g ◦ f. (This means that composition is not commutative).
f ◦ g ◦ h is the composition that composes f with g with h.
Since when we combine functions in composition to make a new function, sometimes we
define a function to be the composition of two smaller function. For instance,
h = f ◦ g (1)
h is the function that is made from f composed with g.
For regular functions such as, say:
f(x) = 3x
2
+ 2x + 1 (2)
What do we end up doing with this function? All we do is plug in various values of x into
the function because that’s what the function accepts as inputs. So we would have different
outputs for each input:
f(−2) = 3(−2)
2
+ 2(−2) + 1 = 12 − 4 + 1 = 9 (3)
f(0) = 3(0)
2
+ 2(0) + 1 = 1 (4)
f(2) = 3(2)
2
+ 2(2) + 1 = 12 + 4 + 1 = 17 (5)
When composing functions we do the same thing but instead of plugging in numbers we are
plugging in whole functions. For example let’s look at the following problems below:
Examples
• Find (f ◦ g)(x) for f and g below.
f(x) = 3x + 4 (6)
g(x) = x
2
+
1
x
(7)
When composing functions we always read from right to left. So, first, we will plug x
into g (which is already done) and then g into f. What this means, is that wherever we
see an x in f we will plug in g. That is, g acts as our new variable and we have f (g(x)).
1
g(x) = x
2
+
1
x
(8)
f(x) = 3x + 4 (9)
f( ) = 3( ) + 4 (10)
f(g(x)) = 3(g(x)) + 4 (11)
f(x
2
+
1
x
) = 3(x
2
+
1
x
) + 4 (12)
f(x
2
+
1
x
) = 3x
2
+
3
x
+ 4 (13)
Thus, (f ◦ g)(x) = f (g(x)) = 3x
2
+
3
x
+ 4.
Let’s try one more composition but this time with 3 functions. It’ll be exactly the same but
with one extra step.
• Find (f ◦ g ◦ h)(x) given f, g, and h below.
f(x) = 2x (14)
g(x) = x
2
+ 2x (15)
h(x) = 2x (16)
(17)
We wish to find f (g(h(x))). We will first find g(h(x)).
h(x) = 2x (18)
g( ) = ( )
2
+ 2( ) (19)
g(h(x)) = (h(x))
2
+ 2(h(x)) (20)
g(2x) = (2x)
2
+ 2(2x) (21)
g(2x) = 4x
2
+ 4x (22)
Thus g(h(x)) = 4x
2
+ 4x. We now wish to find f (g(h(x))).
g(h(x)) = 4x
2
+ 4x (23)
f( ) = 2( ) (24)
f(g(h(x))) = 2(g(h(x))) (25)
f(4x
2
+ 4x) = 2(4x
2
+ 4x) (26)
f(4x
2
+ 4x) = 8x
2
+ 8x (27)
(28)
Thus (f ◦ g ◦ h)(x) = f (g(h(x))) = 8x
2
+ 8x.
2
Here are some example problems for you to work out on your own with their respective
answers at the bottom:
Find (s ◦ p)(x) for f and g below.
s(x) = 4x
2
+ 8x + 8 (29)
p(x) = x + 4 (30)
Find (g ◦ f ◦ q)(t) for g, f, and q below.
q(t) =
√
x (31)
f(t) = x
2
(32)
g(t) = 5x
9
(33)
Find (f ◦ g ◦ h ◦ j)(x) for the functions below. HINT: Look at f and think about what will
happen to it no matter what we plug into f.
j(x) = 4x
9
+ 3sin(x) (34)
h(x) = ln(x) (35)
g(x) = 4x (36)
f(x) = 1 (37)
answers in order: 4x
2
+ 40x + 104, 5t
9
, 1
3
