Composite Functions with Numbers
Composite Functions with Numbers - How it Works - Video
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 - (f ∘ g)(x)
Example 1 (f ∘ g)(x):
Here we want to find (f ∘ g)(x) when x = 3. So our first step is to find when g(x) first since it is the inner function. Then whatever output value we get will be our new input for f(x).
We have two functions f(x) = x2 and g(x) = 3x - 5.
Let's find g(x) first since function g is in the inner function.
g(x) = 3x - 5
g(3) = 3 * (3) - 5
g(3) = 9 - 5
g(3) = 4
Substituted the input value, 3.
Multiplied.
Subtracted like terms.
f(x) = x2
f(4) = (4)2
f(4) = 16
Substituted the new input value, 4.
Squared.
Here we have our answer after substituting twice, f(4) = 16.
Let's check our answer by substituting the equations instead of one value.
(f ∘ g)(x) = f( g(x) )
= f [ (3x -5) ]
= [ (3x - 5) ]2
= (3x - 5) * (3x - 5)
= 9x2 - 15x - 15x + 25
= 9x2 - 30x + 25
Substituted the inner function g(x).
Substituted into the outer function f(x).
Wrote the binomial twice.
Distributed the binomial.
Combined like terms.
(f ∘ g)(x) = 9x2 - 30x + 25
(f ∘ g)(3) = 9 * (3)2 - 30 * (3) + 25
(f ∘ g)(3) = 9 * 9 - 90 + 25
(f ∘ g)(3) = 81 - 90 + 25
(f ∘ g)(3) = 16
Substituted the input value, 3.
Squared and multiplied.
Multiplied.
Combined like terms.
Here we have our answer after finding (f ∘ g)(x) = 9x2 - 30x + 25. We substituted the value in the question and got (f ∘ g)(3) = 16. We also know f(4) = 16 so we can say (f ∘ g)(3) = f( g(3) ) = f(4) = 16.
Example 1 - (g ∘ f)(x)
Example 1 (g ∘ f)(x):
Here we want to find (g ∘ f)(x) when x = 3. So our first step is to find when f(x) first since it is the inner function. Then whatever output value we get will be our new input for g(x).
We have two functions f(x) = x2 and g(x) = 3x - 5.
Let's find f(x) first since function f is in the inner function.
f(x) = x2
f(3) = (3)2
f(3) = 9
Substituted the input value, 3.
Squared.
g(x) = 3x - 5
g(9) = 3 * (9) - 5
g(9) = 27 - 5
g(9) = 22
Substituted the new input value, 9.
Multiplied.
Combined like terms.
Here we have our answer after substituting twice, g(9) = 22.
Let's check our answer by substituting the equations instead of one value.
(g ∘ f)(x) = g( f(x) )
= g [ (x2) ]
= 3 * (x2) - 5
= 3 * x2 - 5
Substituted the inner function f(x).
Substituted into the outer function g(x).
Dropped the parentheses.
(g ∘ f)(x) = 3 * x2 - 5
(g ∘ f)(3) = 3 * (3)2 - 5
(g ∘ f)(3) = 3 * 9 - 5
(g ∘ f)(3) = 27 - 5
(g ∘ f)(3) = 22
Substituted the input value, 3.
Squared.
Multiplied.
Combined like terms.
Here we have our answer after finding (g ∘ f)(x) = 3 * x2 - 5. We substituted the value in the question and got (g ∘ f)(3) = 22. We also know g(9) = 22 so we can say (g ∘ f)(3) =g( f(3) ) = g(9) = 22.