Composite Functions
If you understood what functions and inverse functions are, you will definitely understand what composite functions are.
Lets have two functions
f(x) = 2x + 4
g(x) = 3x + 25
So f(x) is one function and g(x) is another function. We have used g(x) for the notation OF ANOTHER FUNCTION. So we have two functions all together.
Lets say we wanted to know what g(2) equals and what f(4) equals; the working is shown below.
g(2) = 3(2) + 25 = 6 + 25 = 31
f(4) = 2(4) + 4 = 8 + 4 = 12
Question: What is the value of f(g(2))?
Look at this carefully, this is not hard to understand. In f(x), the x has been replaced with g(2). So we have inputted g(2) into the function f(x). But what does g(2) equal? g(2) = 31
SO
We can write f(g(2)) = f(31)
f(31) = 2(31) + 4 = 62 + 4 = 66
So f(g(2)) = 66
What does g(f(4)) equal?
Look at this carefully, this is not hard to understand. In g(x), the x has been replaced with f(4). So we have inputted f(4) into the function g(x). But what does f(4) equal? f(4) = 12
SO
We can write g(f(4)) = g(12)
g(12) = 3(12) + 25 = 36 + 25 = 61
So g(f(4)) = 61
Taking forward the same two functions, which are written below, lets find what f(g(x)) equals.
f(x) = 2x + 4 g(x) = 3x + 25
What does f(g(x)) mean first? NOTHING HAS CHANGED, USE THE SAME IDEA AND METHOD APPLIED ABOVE.
The function g(x) is an input in the function f(x). So we have replaced the x in the function f(x) with the function g(x). What does g(x) equal though? We know what g(x) equals, as we know what both functions equal, g(x) = 3x + 25, as written below.
So f(g(x)) = f(3x + 25)
Given f(x) = 2x + 4, f(3x + 25) would mean that the expression 3x + 25 would be inputted into the function f(x).
So if f(x) = 2x + 4,
then f(3x + 25) = 2(3x + 25) + 4 = 6x + 50 + 4 = 6x + 54
We have substituted the x in 2x + 3 with the expression (3x + 4), as that is the input.
So
f(g(x)) = f(3x + 25) = 2(3x + 25) + 4 = 6x + 54
We can simply write that
f(g(x)) = 6x + 54 THIS IS WHAT YOU CALL A COMPOSITE FUNCTION
The composite function tells us that if we want to input into f(x) the output of the function g(x) (READ THIS SENTENCE AGAIN, IF YOU DO NOT UNDERSTAND), we can attain the output of the function f(x) by using the composite function.
f(g(x)) = 6x + 54
Remember when we wanted to find f(g(2)), we first found what g(2) equalled, then we inputted what g(2) equalled into the function f(x), but we could of found the value of f(g(2)) using the composite function, as shown below.
f(g(x)) = 6x + 54
If x = 2, then f(g(2)) = 6(2) + 54 = 12 + 54 = 66 (THIS IS THE SAME ANSWER AS SHOWN ABOVE, GO CHECK)
You make ask why need composite functions when we can work what g(2) equals first and then input that answer into the function f(x). However, by making and using a composite function, it is much quicker.
Lets now find g(f(x))
f(x) = 2x + 4 g(x) = 3x + 25
f(x) is the input in the function g(x), so we have replaced the x in the function g(x) with f(x). But what does f(x) equal though? We know what f(x) equals, as we know what both functions are equal to, f(x) = 2x + 4.
So g(f(x)) = g(2x + 4)
Given that g(x) = 3x + 25, then g(2x + 4) would mean that the expression (2x + 4) would be inputted into the function g(x).
So if g(x) = 3x + 25,
then g(2x + 4) = 3(2x + 4) + 25 = 6x + 12 + 25 = 6x + 37
We have substituted the x in 3x + 25 with the expression (2x + 4), as that is the input.
So
g(f(x)) = g(2x + 4) = 3(2x + 4) + 25 = 6x + 37
We can simply write
g(f(x)) = 6x + 37 (THIS IS WHAT YOU CALL A COMPOSITE FUNCTION)
The composite function tells us that if we want to input into g(x) the output of the function f(x) (READ THIS SENTENCE SLOWLY AGAIN, IF YOU DO NOT UNDERSTAND), we can attain the output of the function g(x) by using the composite function.
Remember when we wanted to find g(f(4)), we first found what f(4) equalled, then we inputted what f(4) equalled into the function g(x), but we could of found the value of g(f(4)) using the composite function, as shown below.
g(f(4)) = 6(4) + 37 = 24 + 37 = 61 (THIS IS THE SAME ANSWER AS SHOWN ABOVE, GO CHECK)
So now you know the importance of composite functions and how you can make them.
Lets have two functions
f(x) = 2x + 4
g(x) = 3x + 25
So f(x) is one function and g(x) is another function. We have used g(x) for the notation OF ANOTHER FUNCTION. So we have two functions all together.
Lets say we wanted to know what g(2) equals and what f(4) equals; the working is shown below.
g(2) = 3(2) + 25 = 6 + 25 = 31
f(4) = 2(4) + 4 = 8 + 4 = 12
Question: What is the value of f(g(2))?
Look at this carefully, this is not hard to understand. In f(x), the x has been replaced with g(2). So we have inputted g(2) into the function f(x). But what does g(2) equal? g(2) = 31
SO
We can write f(g(2)) = f(31)
f(31) = 2(31) + 4 = 62 + 4 = 66
So f(g(2)) = 66
What does g(f(4)) equal?
Look at this carefully, this is not hard to understand. In g(x), the x has been replaced with f(4). So we have inputted f(4) into the function g(x). But what does f(4) equal? f(4) = 12
SO
We can write g(f(4)) = g(12)
g(12) = 3(12) + 25 = 36 + 25 = 61
So g(f(4)) = 61
Taking forward the same two functions, which are written below, lets find what f(g(x)) equals.
f(x) = 2x + 4 g(x) = 3x + 25
What does f(g(x)) mean first? NOTHING HAS CHANGED, USE THE SAME IDEA AND METHOD APPLIED ABOVE.
The function g(x) is an input in the function f(x). So we have replaced the x in the function f(x) with the function g(x). What does g(x) equal though? We know what g(x) equals, as we know what both functions equal, g(x) = 3x + 25, as written below.
So f(g(x)) = f(3x + 25)
Given f(x) = 2x + 4, f(3x + 25) would mean that the expression 3x + 25 would be inputted into the function f(x).
So if f(x) = 2x + 4,
then f(3x + 25) = 2(3x + 25) + 4 = 6x + 50 + 4 = 6x + 54
We have substituted the x in 2x + 3 with the expression (3x + 4), as that is the input.
So
f(g(x)) = f(3x + 25) = 2(3x + 25) + 4 = 6x + 54
We can simply write that
f(g(x)) = 6x + 54 THIS IS WHAT YOU CALL A COMPOSITE FUNCTION
The composite function tells us that if we want to input into f(x) the output of the function g(x) (READ THIS SENTENCE AGAIN, IF YOU DO NOT UNDERSTAND), we can attain the output of the function f(x) by using the composite function.
f(g(x)) = 6x + 54
Remember when we wanted to find f(g(2)), we first found what g(2) equalled, then we inputted what g(2) equalled into the function f(x), but we could of found the value of f(g(2)) using the composite function, as shown below.
f(g(x)) = 6x + 54
If x = 2, then f(g(2)) = 6(2) + 54 = 12 + 54 = 66 (THIS IS THE SAME ANSWER AS SHOWN ABOVE, GO CHECK)
You make ask why need composite functions when we can work what g(2) equals first and then input that answer into the function f(x). However, by making and using a composite function, it is much quicker.
Lets now find g(f(x))
f(x) = 2x + 4 g(x) = 3x + 25
f(x) is the input in the function g(x), so we have replaced the x in the function g(x) with f(x). But what does f(x) equal though? We know what f(x) equals, as we know what both functions are equal to, f(x) = 2x + 4.
So g(f(x)) = g(2x + 4)
Given that g(x) = 3x + 25, then g(2x + 4) would mean that the expression (2x + 4) would be inputted into the function g(x).
So if g(x) = 3x + 25,
then g(2x + 4) = 3(2x + 4) + 25 = 6x + 12 + 25 = 6x + 37
We have substituted the x in 3x + 25 with the expression (2x + 4), as that is the input.
So
g(f(x)) = g(2x + 4) = 3(2x + 4) + 25 = 6x + 37
We can simply write
g(f(x)) = 6x + 37 (THIS IS WHAT YOU CALL A COMPOSITE FUNCTION)
The composite function tells us that if we want to input into g(x) the output of the function f(x) (READ THIS SENTENCE SLOWLY AGAIN, IF YOU DO NOT UNDERSTAND), we can attain the output of the function g(x) by using the composite function.
Remember when we wanted to find g(f(4)), we first found what f(4) equalled, then we inputted what f(4) equalled into the function g(x), but we could of found the value of g(f(4)) using the composite function, as shown below.
g(f(4)) = 6(4) + 37 = 24 + 37 = 61 (THIS IS THE SAME ANSWER AS SHOWN ABOVE, GO CHECK)
So now you know the importance of composite functions and how you can make them.