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Composition of Functions |
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Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.
Using descriptive variables, we can notate these two functions. The function C( T )
gives the cost C
of heating a house for a given average daily temperature in T
degrees Celsius. The function T( d )
gives the average daily temperature on day d
of the year. For any given day, Cost=C( T( d ) )
means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature T( d ).
For example, we could evaluate T( 5 )
to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function{: data-type="term" .no-emphasis} at that temperature. We would write C( T( 5 ) ).
By combining these two relationships into one function, we have performed function composition, which is the focus of this section.
Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If w(y)
is the wife’s income and h(y)
is the husband’s income in year y,
and we want T
to represent the total income, then we can define a new function.
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functions f( x )
and g( x )
with real number outputs, we define new functions f+g, f−g, fg,
and f g
by the relations
and ( g f )( x ),
given f( x )=x−1
and g( x )= x 2 −1.
Are they the same function?
Note: For ( g f )( x ),
the condition x≠1
is necessary because when x=1,
the denominator is equal to 0, which makes the function undefined.
and ( f−g )( x ).
No, the functions are not the same.
Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions{: data-type="term" .no-emphasis} so that the output of one function becomes the input of another is known as a composition of functions{: data-type="term" .no-emphasis}. The resulting function is known as a composite function. We represent this combination by the following notation:
We read the left-hand side as “f
composed with g
at x,”
and the right-hand side as “f
of g
of x.”
The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol ∘
is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases f(g(x))≠f(x)g(x).
It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function g
takes the input x
first and yields an output g( x ).
Then the function f
takes g( x )
as an input and yields an output f( g( x ) ).
In general, f∘g
and g∘f
are different functions. In other words, in many cases f( g( x ) )≠g( f( x ) )
for all x.
We will also see that sometimes two functions can be composed only in one specific order.
For example, if f( x )= x 2
and g( x )=x+2,
then
but
These expressions are not equal for all values of x,
so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value x=− 1 2 .
Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.
and functions f
and g,
this action defines a composite function{: data-type="term"}, which we write as f∘g
such that
is all x
such that x
is in the domain of g
and g( x )
is in the domain of f.
It is important to realize that the product of functions fg
is not the same as the function composition f( g( x ) ),
because, in general, f( x )g( x )≠f( g( x ) ).
and g( f( x ) ).
Determine whether the composition of the functions is commutative{: data-type="term" .no-emphasis}.
into f( x ).
into g( x ).
so the operation of function composition is not commutative.
gives the number of calories burned completing s
sit-ups, and s(t)
gives the number of sit-ups a person can complete in t
minutes. Interpret c(s(3)).
Because the input to the s-function is time, t=3
represents 3 minutes, and s(3)
is the number of sit-ups completed in 3 minutes.
Using s(3)
as the input to the function c(s)
gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).
gives miles that can be driven in x
hours and g(y)
gives the gallons of gas used in driving y
miles. Which of these expressions is meaningful: f( g(y) )
or g( f(x) )?
is a function whose output is the number of miles driven corresponding to the number of hours driven.
is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:
takes miles as the input and a number of gallons as the output. The function f(x)
requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression f( g(y) )
is meaningless.
The expression f(x)
takes hours as input and a number of miles driven as the output. The function g(y)
requires a number of miles as the input. Using f(x)
(miles driven) as an input value for g(y),
where gallons of gas depends on miles driven, does make sense. The expression g( f(x) )
makes sense, and will yield the number of gallons of gas used, g,
driving a certain number of miles, f(x),
in x
hours.
The acceleration of a planet subjected to any force F
is given by the function a(F).
Form a meaningful composition of these two functions, and explain what it means.
makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G( a(F) )
does not make sense.
Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.
When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.
and g(f(3)).
| x | f(x) | g(x) |
|---|---|---|
| 1 | 6 | 3 |
| 2 | 8 | 5 |
| 3 | 3 | 2 |
| 4 | 1 | 7 |
we start from the inside with the input value 3. We then evaluate the inside expression g(3)
using the table that defines the function g:
g(3)=2.
We can then use that result as the input to the function f,
so g(3)
is replaced by 2 and we get f(2).
Then, using the table that defines the function f,
we find that f(2)=8.
we first evaluate the inside expression f(3)
using the first table: f(3)=3.
Then, using the table for g,
we can evaluate
and g∘f
as tables.
| x | g( x ) | f( g( x ) ) | f( x ) | g( f( x ) ) |
| 3 | 2 | 8 | 3 | 2 |
and g(f(4)).
and g(f(4))=g(1)=3
When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the x-
and y-
axes of the graphs.
-
Locate the given input to the inner function on the x-
axis of its graph.
-
Read off the output of the inner function from the y-
axis of its graph.
-
Locate the inner function output on the x-
axis of the graph of the outer function.
-
Read the output of the outer function from the y-
axis of its graph. This is the output of the composite function. {: type="1"}
{: #Figure_01_04_002}
we start with the inside evaluation. See [link].
{: #Figure_01_04_004}
We evaluate g(1)
using the graph of g(x),
finding the input of 1 on the x-
axis and finding the output value of the graph at that input. Here, g(1)=3.
We use this value as the input to the function f.
finding the input of 3 on the x-
axis and reading the output value of the graph at this input. Here, f(3)=6,
so f(g(1))=6.
{: #Figure_01_04_005}
When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition f( g( x ) ).
To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like f(t)= t 2 −t,
we substitute the value inside the parentheses into the formula wherever we see the input variable.
- Evaluate the inside function using the input value or variable provided.
- Use the resulting output as the input to the outside function. {: type="1"}
and h(x)=3x+2,
evaluate f(h(1)).
we start by evaluating h(x)
at 1.
so we evaluate f(t)
at an input of 5.
and x
were called in this problem because we evaluated for specific numerical values.
and h(x)=3x+2,
evaluate
-
h(f(2))
-
h(f(−2)) {: type="a"}
As we discussed previously, the domain of a composite function{: data-type="term" .no-emphasis} such as f∘g
is dependent on the domain of g
and the domain of f.
It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as f∘g.
Let us assume we know the domains of the functions f
and g
separately. If we write the composite function for an input x
as f( g( x ) ),
we can see right away that x
must be a member of the domain of g
in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that g( x )
must be a member of the domain of f,
otherwise the second function evaluation in f( g( x ) )
cannot be completed, and the expression is still undefined. Thus the domain of f∘g
consists of only those inputs in the domain of g
that produce outputs from g
belonging to the domain of f.
Note that the domain of f
composed with g
is the set of all x
such that x
is in the domain of g
and g( x )
is in the domain of f.
is the set of those inputs x
in the domain of g
for which g( x )
is in the domain of f.
determine its domain.**
-
Find the domain of g.
-
Find the domain of f.
-
Find those inputs x
in the domain of g
for which g( x )
is in the domain of f.
That is, exclude those inputs x
from the domain of g
for which g( x )
is not in the domain of f.
The resulting set is the domain of f∘g. {: type="1"}
consists of all real numbers except x= 2 3 ,
since that input value would cause us to divide by 0. Likewise, the domain of f
consists of all real numbers except 1. So we need to exclude from the domain of g( x )
that value of x
for which g( x )=1.
is the set of all real numbers except 2 3
and 2.
This means that
is ( −∞,3 ].
Now we check the domain of the composite function
since the radicand of a square root must be positive. Since square roots are positive, 3−x ≥0 ,
or, 3−x ≥0,
which gives a domain of (-∞,3]
.
can contain values that are not in the domain of f,
though they must be in the domain of g.
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function{: data-type="term" .no-emphasis}, so we may choose the decomposition that appears to be most expedient.
as the composition of two functions.
and h,
so f(x)=g(h(x)).
To do this, we look for a function inside a function in the formula for f(x).
As one possibility, we might notice that the expression 5− x 2
is the inside of the square root. We could then decompose the function as
as the composition of two functions.
g( x )= 4+ x 2
{: data-type="newline"}
h( x )= 4 3−x
{: data-type="newline"}
f=h∘g
| Composite function | ( f∘g )( x )=f( g( x ) ) |
- We can perform algebraic operations on functions. See [link].
- When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
- The function produced by combining two functions is a composite function. See [link] and [link].
- The order of function composition must be considered when interpreting the meaning of composite functions. See [link].
- A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
- A composite function can be evaluated from a table. See [link].
- A composite function can be evaluated from a graph. See [link].
- A composite function can be evaluated from a formula. See [link].
- The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See [link] and [link].
- Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
- Functions can often be decomposed in more than one way. See [link].
equal to zero, and check for any other domain restrictions on f
and g,
such as an even-indexed root or zeros in the denominator.
Then f(g(x))=f(x−1)=(x−1)+1=x
and g(f(x))=g(x+1)=(x+1)−1=x.
So f∘g=g∘f.
and g(x)=6− x 2 ,
find f+g, f−g, fg,
and f g .
Determine the domain for each function in interval notation.
domain: (−∞,∞)
(f−g)( x )=2 x 2 +2x−6,
domain: (−∞,∞)
(fg)( x )=− x 4 −2 x 3 +6 x 2 +12x,
domain: (−∞,∞)
( f g )( x )= x 2 +2x 6− x 2 ,
domain: (−∞,− 6 )∪(− 6 , 6 )∪( 6 ,∞)
and g(x)=5,
find f+g, f−g, fg,
and f g .
Determine the domain for each function in interval notation.
and g(x)= 1 2x ,
find f+g, f−g, fg,
and f g .
Determine the domain for each function in interval notation.
domain: (−∞,0)∪(0,∞)
(f−g)( x )= 4 x 3 +8 x 2 −1 2x ,
domain: (−∞,0)∪(0,∞)
(fg)( x )=x+2,
domain: (−∞,0)∪(0,∞)
( f g )( x )=4 x 3 +8 x 2 ,
domain: (−∞,0)∪(0,∞)
and g(x)= 1 6−x ,
find f+g, f−g, fg,
and f g .
Determine the domain for each function in interval notation.
and g(x)= x−5 ,
find f+g, f−g, fg,
and f g .
Determine the domain for each function in interval notation.
domain: [5,∞)
(f−g)(x)=3 x 2 − x−5 ,
domain: [5,∞)
(fg)(x)=3 x 2 x−5 ,
domain: [5,∞)
( f g )(x)= 3 x 2 x−5 ,
domain: (5,∞)
and g(x)=|x−3|,
find g f .
Determine the domain of the function in interval notation.
and g(x)=3x−5,
find the following:
-
f(g(2))
-
f(g(x))
-
g(f(x))
-
( g∘g )( x )
-
( f∘f )( −2 ) {: type="a"}
c. f( g( x ) )=6 x 2 −2;
d. ( g∘g )(x)=3(3x−5)−5=9x−20;
e. ( f∘f )( −2 )=163
For the following exercises, use each pair of functions to find f( g( x ) )
and g( f( x ) ).
Simplify your answers.
For the following exercises, use each set of functions to find f( g( h(x) ) ).
Simplify your answers.
g(x)=x−6,
and h(x)= x
g(x)= 1 x ,
and h(x)=x+3
and g(x)=x−3,
find the following:
-
(f∘g)(x)
-
the domain of (f∘g)(x)
in interval notation
-
(g∘f)(x)
-
the domain of (g∘f)(x)
-
( f g )x {: type="a"}
and g(x)=− 3 x ,
find the following:
-
(g∘f)(x)
-
the domain of (g∘f)(x)
in interval notation {: type="a"}
b. ( −∞, 1 2 )
find the following:
-
(g∘f)(x)
-
(g∘f)(2) {: type="a"}
and m(x)= x 2 −4,
state the domain of each of the following functions using interval notation:
-
p(x) m(x)
-
p(m(x))
-
m(p(x)) {: type="a"}
b. (−∞,−2)∪(2,∞);
c. (0,∞)
and h(x)= x 2 −9,
state the domain of each of the following functions using interval notation.
-
q(x) h(x)
-
q( h(x) )
-
h( q(x) ) {: type="a"}
and g(x)= x−1 ,
write the domain of (f∘g)(x)
in interval notation.
For the following exercises, find functions f(x)
and g(x)
so the given function can be expressed as h(x)=f( g(x) ).
{: data-type="newline"}
g(x)=2x+6
{: data-type="newline"}
g(x)=(x−1)
{: data-type="newline"}
g(x)= 1 x−2
{: data-type="newline"}
g(x)= 2x−1 3x+4
For the following exercises, use the graphs of f,
shown in [link], and g,
shown in [link], to evaluate the expressions.
{: #Figure_01_04_201}
{: #Figure_01_04_202}
For the following exercises, use graphs of f(x),
shown in [link], g(x),
shown in [link], and h(x),
shown in [link], to evaluate the expressions.
{: #Figure_01_04_203}
{: #Figure_01_04_204}
{: #Figure_01_04_205}
For the following exercises, use the function values for f and g
shown in [link] to evaluate each expression.
| x | f(x) | g(x) | | 0 | 7 | 9 | | 1 | 6 | 5 | | 2 | 5 | 6 | | 3 | 8 | 2 | | 4 | 4 | 1 | | 5 | 0 | 8 | | 6 | 2 | 7 | | 7 | 1 | 3 | | 8 | 9 | 4 | | 9 | 3 | 0 | {: #Table_01_04_03 summary="Eleven columns and three rows. The first row is labeled, “x”, the second row is labeled, “f(x)”, and the row column is labeled, “g(x)”. We have the following values for f(x): f(0)=7, f(1)=6, f(2)=5, f(3)=8, f(4)=4, f(5)=0, f(6)=2, f(7)=1, f(8)=9, and f(9)=3. And for g(0)=9, g(1)=5, g(2)=6, g(3)=2, g(4)=1, g(5)=8, g(6)=7, g(7)=3, g(8)=4, and g(9)=0."}
For the following exercises, use the function values for f and g
shown in [link] to evaluate the expressions.
| **x
** | f(x) | g(x) | | -3 | 11 | -8 | | -2 | 9 | -3 | | -1 | 7 | 0 | | 0 | 5 | 1 | | 1 | 3 | 0 | | 2 | 1 | -3 | | 3 | -1 | -8 | {: #Table_01_04_04 summary="Three columns and eight rows. The first column is labeled, “x”, the second column is labeled, “f(x)”, and the third column is labeled, “g(x)”. We have the following values for f(x): f(-3)=11, f(-2)=9, f(-1)=7, f(0)=5, f(1)=3, f(2)=1, and f(3)=-1. And for g(-3)=-8, g(-2)=-3, g(-1)=0, g(0)=1, g(1)=0, g(2)=-3, and g(3)=-8."}
For the following exercises, use each pair of functions to find f( g( 0 ) )
and g( f(0) ).
For the following exercises, use the functions f(x)=2 x 2 +1
and g(x)=3x+5
to evaluate or find the composite function as indicated.
For the following exercises, use f(x)= x 3 +1
and g(x)= x−1 3 .
and (g∘f)(x).
Compare the two answers.
and (g∘f)(2).
-
Find (f∘f)(x).
-
Is (f∘f)(x)
for any function f
the same result as the answer to part (a) for any function? Explain. {: type="a"}
For the following exercises, let F(x)= (x+1) 5 ,
f(x)= x 5 ,
and g(x)=x+1.
For the following exercises, find the composition when f(x)= x 2 +2
for all x≥0
and g(x)= x−2 .
; (g∘f)(6)=6
gives the number of items that will be demanded when the price is p.
The production cost C(x)
is the cost of producing x
items. To determine the cost of production when the price is $6, you would do which of the following?
-
Evaluate D( C(6) ).
-
Evaluate C( D(6) ).
-
Solve D( C(x) )=6.
-
Solve C( D(p) )=6. {: type="a"}
gives the pain level on a scale of 0 to 10 experienced by a patient with d
milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after t
minutes is modeled by m(t).
Which of the following would you do in order to determine when the patient will be at a pain level of 4?
-
Evaluate A( m(4) ).
-
Evaluate m( A(4) ).
-
Solve A( m(t) )=4.
-
Solve m( A(d) )=4. {: type="a"}
of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function P(x)
that computes the final price of the item in terms of the original price x.
(Hint: Use function composition to find your answer.)
find the area of the ripple as a function of time. Find the area of the ripple at t=2.
and A(2)=π ( 25 4 ) 2 =2500π
square inches
express the area burned as a function of time, t
(minutes).
square units
in inches, of a spherical balloon is related to the volume, V,
by r(V)= 3V 4π 3 .
Air is pumped into the balloon, so the volume after t
seconds is given by V(t)=10+20t.
-
Find the composite function r( V(t) ).
-
Find the exact time when the radius reaches 10 inches. {: type="a"}
3<T<33,
where T
is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by T(t)=5t+1.5,
where t
is the time in hours.
-
Find the composite function N( T(t) ).
-
Find the time (round to two decimal places) when the bacteria count reaches 6752. {: type="a"}
b. 3.38 hours
composite function : the new function formed by function composition, when the output of one function is used as the input of another