Embedded Files
Composite Functions with Functions - How it Works - Video
Composite Functions with Functions - How it Works - Video
Definitions
Definitions
Definition:
Here we have the definition of composite functions. With composite functions, it is like double substitution. The key here is to work inside out so start in the middle and work outward.
One that often takes a back pedal is domain of a composite function. The domain of f and g is not always in your answer of (f ∘ g) or (g ∘ f). You should not take the domain after simplifying. That is where people make mistakes. Sometimes, you get lucky and it is the same, but not often. You should find the domain right after substituting the inside function into the outer function.
Example 1
Example 1
Example 1:
We have to two functions f(x) = 3x -1 and g(x) = x2 - 2.
On the left we (f ∘ g)(x) = f( g[x] ). Remember we work from the inside out or from the middle outward. So our first step is to substitute g[x] into what we have so far and then follow order of operations.
On the right we (g ∘ f)(x) = g( f[x] ). Remember we work from the inside out or from the middle outward. So our first step is to substitute f[x] into what we have so far and then follow order of operations.
(f ∘ g)(x) = f (g[x])
= f ( [x2 - 2] )
= 3 * [x2 - 2] - 1
= [3x2 - 6] -1
= 3x2 - 7
Substituted g(x).
Substituted our new input, g(x), into f(x).
Distributed the 3 into the brackets.
Combined like terms.
(g ∘ f)(x) = g (f[x])
= g ( [3x - 1] )
= (3x -1)2 - 2
= (3x -1) * (3x -1) - 2
= (9x2 - 6x + 1) - 2
= 9x2 - 6x - 1
Substituted f(x).
Substituted our new input, f(x), into g(x).
Separated the squared into 2 terms.
Distributed the binomials.
Combined like terms.
Example 1 - Check
Example 1 - Check
Example 1 - Check:
Here we show two ways to find (f ∘ g)(-2) and (g ∘ f)(-2). Hopefully they are both the same.
g(x) = x2 - 2
g(-2) = (-2)2 - 2
g(-2) = 4 - 2
g(-2) = 2
Substituted x = -2 into g(x).
Squared the term.
Subtracted like terms.
f(x) = 3x - 1
f(2) = 3*(2) - 1
f(2) = 6 -1
f(2) = 5
Substituted x= 2 into f(x).
Multiplied terms.
Subtracted like terms.
We have to be careful here. When we substitute in each function separately, we substitute the number we want x = -2 into the first one. Once we find that output, that output is our new input for the second equation. We do not use the same input as our first equation unless the output of the first function is the same.
So final answer is 5.
(f ∘ g)(x) = 3x2 - 7
(f ∘ g)(-2) = 3* (-2)2 - 7
(f ∘ g)(-2) = 3* 4 - 7
(f ∘ g)(-2) = 12 - 7
(f ∘ g)(-2) = 5
Substituted x = -2 into f(x).
Squared.
Multiplied.
Subtracted like terms.
We have our final is (f ∘ g)(-2) = 5, which is the same as substituting into the two different functions.
f(x) = 3x - 1
f(-2) = 3*(-2) - 1
f(2) = -6 -1
f(-2) = -7
Substituted x= -2 into f(x).
Multiplied terms.
Subtracted like terms.
g(x) = x2 - 2
g(-7) = (-7)2 - 2
g(-7) = 49 - 2
g(-2) = 47
Substituted x= -2 into g(x).
Squared the term.
Subtracted like terms.
We have to be careful here. When we substitute in each function separately, we substitute the number we want x = -2 into the first one. Once we find that output, that output is our new input for the second equation. We do not use the same input as our first equation unless the output of the first function is the same.
So final answer is 47.
(g ∘ f)(x) = 9x2 - 6x - 1
(g ∘ f)(-2) = 9 * (-2)2 - 6 * (-2) - 1
(g ∘ f)(-2) = 9 * 4 - [-12] - 1
(g ∘ f)(-2) = 36 + 12 - 1
(g ∘ f)(-2) = 47
Substituted x = -2 into f(x).
Multiplied two different terms.
Multiplied and changed sign.
Added like terms.
We have our final is (g ∘ f)(-2) = 47, which is the same as substituting into the two different functions.
Live Worksheet
Live Worksheet
Page updated
Google Sites
Report abuse