Composite Function
How to solve the composite function: definition, 4 examples, and their solutions.
Definition
(g ∘ f)(x) = g(f(x))
To find g(f(x)),
put f(x)
into g( ).
Function
Example(g ∘ f)(x)
(g ∘ f)(x) = g(f(x))
Solve this from the inside.
f(x) = 3x
So g(f(x)) = g(3x).
g(x) = x2 - x - 1
To solve g(3x),
put [3x]
into g( ) = [ ]2 - [ ] - 1..
Then g(3x) = [3x]2 - [3x] - 1.
[3x]2
= 32x2
= 9x2
Power of a Product
So (g ∘ f)(x) = 9x2 - 3x - 1.
Example(f ∘ g)(x)
Compare (f ∘ g)(x) = 3x2 - 3x - 3
and the previous answer (g ∘ f)(x) = 9x2 - 3x - 1.
As you can see,
(f ∘ g)(x) ≠ (g ∘ f)(x).
(f ∘ g)(x) = f(g(x))
Solve this from the inside.
g(x) = x2 - x - 1
So f(g(x)) = f(x2 - x - 1).
f(x) = 3x
To solve f(x2 - x - 1),
put [x2 - x - 1]
into f( ) = 3[ ].
Then f(x2 - x - 1) = 3[x2 - x - 1].
3[x2 - x - 1]
= 3x2 - 3x - 3
Multiply a Monomial and a Polynomial
So (f ∘ g)(x) = 3x2 - 3x - 3.
Compare (f ∘ g)(x) = 3x2 - 3x - 3
and the previous answer (g ∘ f)(x) = 9x2 - 3x - 1.
As you can see,
(f ∘ g)(x) ≠ (g ∘ f)(x).
Example(f ∘ f)(x)
(f ∘ f)(x) = f(f(x))
Solve this from the inside.
f(x) = 2x - 1
So f(f(x)) = f(2x - 1).
f(x) = 2x - 1
To solve f(2x - 1),
put [2x - 1]
into f( ) = 2[ ] - 1.
Then f(2x - 1) = 2[2x - 1] - 1.
2[2x - 1] = 4x - 2
-2 - 1 = -3
So (f ∘ f)(x) = 4x - 3.
Example(f ∘ f ∘ f)(x)
(f ∘ f ∘ f)(x) = f(f(f(x)))
Solve this from the inside.
f(x) = 3x + 1
So f(f(f(x))) = f(f(3x + 1)).
f(x) = 3x + 1
To solve f(3x + 1),
put [3x + 1]
into f(f( )) = f(3[ ] + 1).
Then f(f(3x + 1)) = f(3[3x + 1] + 1).
3[3x + 1] = 9x + 3
+3 + 1 = +4
f(x) = 3x + 1
To solve f(9x + 4),
put [9x + 4]
into f( ) = 3[ ] + 1.
Then f(9x + 4) = 3[9x + 4] + 1.
3[9x + 4] = 27x + 12
+12 + 1 = +13
So (f ∘ f ∘ f)(x) = 27x + 13.