Answer/Step-by-step explanation:
Composition functions are functions that combine to make a new function. We use the notation ◦ 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)).
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.
