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Once a relation is known to be a function, it is handy to write it in a way that directly gives essential information such as the independent and dependent variables. This is where function notation comes to the rescue. This lesson will explore some applications of function notation and how this way of representing functions helps combine different functions.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Evaluating Functions at Another Function
As with algebraic expressions, functions can also be evaluated at a specific input. In the following applet, drag a function into each empty box to explore the outputs obtained when a function is evaluated at another function.
It can be seen that f(g(x))=g(f(x)), which suggests that the order does not affect the result. Is it always true? Is it true for f(h(x)) and h(f(x))?
Discussion
Representing Functions Using Function Notation
There are different ways to represent a function — using tables of values, mapping diagrams, graphs, and equations. However, aside from these, one of the most common ways is using function notation.
Discussion
Evaluating Functions and Finding a Certain Input
In the same way that an expression can be evaluated at a particular x-value, functions can also be evaluated at a specific input. Furthermore, it is also possible to determine the input that produces a specific output.
Example
Modeling With Functions
For the last few days of the long holiday, Izabella and Emily visited a city they did not know. Before returning home, they went into a souvenir shop where all the souvenirs were priced the same. After picking out some gifts and just before paying, Emily found a $3 gift card in her purse.
The function C(s)=4.5s−3 represents the cost, in dollars, of buying s souvenirs with a $3 gift card.
a How much will the girls pay if they buy ten souvenirs?
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b How many souvenirs can the girls buy with $69?
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Hint
b Find the value of s for which the output is 69. Substitute 69 for C(s) and solve the resulting equation for s.
Solution
a Based on the given information, finding how much will the girls pay for buying ten souvenirs is equivalent to finding C(10). This can be done by substituting 10 for s into the function rule and simplifying the right-hand side.
b In this context, C(s) gives the cost of buying s souvenirs. Therefore, asking how many souvenirs the girls can buy with $69 is the same as asking for which value of s the output of C is equal to 69.
C(s)=69
To find such value of s, in the function rule, substitute 69 for C(s) and solve the resulting equation for s.
C(s)=4.5s−3
Substitute
C(s)=69
69=4.5s−3
Solve for s
AddEqn
LHS+3=RHS+3
72=4.5s
DivEqn
LHS/4.5=RHS/4.5
4.572=s
CalcQuot
Calculate quotient
16=s
RearrangeEqn
Rearrange equation
s=16
Pop Quiz
Evaluating Different Functions
Discussion
Combining Functions
Two functions f(x) and g(x) can be combined into a sum, a difference, a product, or a quotient through the following formulas.
(f+g)(x)(f−g)(x)(f⋅g)(x)(gf)(x)=f(x)+g(x)=f(x)−g(x)=f(x)⋅g(x)=g(x)f(x),g(x)=0
However, apart from these four operations, two functions can also be combined into a composition. Concept
Composite Function
A composite function, or a composition of functions, combines two or more functions, which produces a new function. In a composition, the outputs produced by one function are the inputs of the other function. The composition of the functions f and g is denoted as f(g(x)) or (f∘g)(x).
Performing the composition of two functions is similar to evaluating one function into the other. For example, let f(x)=2x+1 and g(x)=x−5. To find f(g(x)), the variable x in f(x)=2x+1 must be substituted with g(x).
Note that f(g(x)) makes sense only when the outputs of g belong to the domain of f. Also, be aware that the composition of functions is not commutative — that is, in general, f(g(x))=g(f(x)). This can be checked with the same two functions.
Just as changing the order of the machines in a factory could alter the final product, changing the order in which the functions are applied could produce different outputs. For example, here f(g(2)) and g(f(2)) are different values.
f(x)=2x+1
Substitute g(x) for x and evaluate
Substitute
x=g(x)
f(g(x))=2g(x)+1
Substitute
f(g(x))=h(x)
h(x)=2g(x)+1
Substitute
g(x)=x−5
h(x)=2(x−5)+1
Distr
Distribute 2
h(x)=2x−10+1
AddTerms
Add terms
h(x)=2x−9
Extra
Parts of a CompositionExplore the parts of a composition by moving the magnifying glass.
Example
Applying Discounts Through Compositions of Functions
Izabella and Emily are shopping for pants. Both girls have a $3 discount coupon for the store. After choosing a pair of pants each, they discovered that the pants are 25% off the marked price. The girls could not agree which order of discount would save them more money, so the cashier allowed them to apply the discounts in whichever order they wanted.
The function f(x)=x−3 represents the effect of applying only the coupon to the marked price, while g(x)=43x represents the effect of applying only the store's sale discount to the marked price. In both functions, x represents the marked price.
a Write a function I(x) that represents how much Izabella will pay if she uses the coupon first and then applies the store's sale discount.
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b Write a function E(x) that represents how much Emily will pay if she uses the store's discount first and then applies the coupon.
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c Which of them saved more money?
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Hint
a Applying the coupon first means to apply f to the marked price. Then, g must be applied to the resulting output. Therefore, I(x) is equal to g(f(x)). To find how much Izabella paid, evaluate I(31).
b This time, apply the store's discount first — that is, start by applying g to the marked price. Then, apply f to the resulting output. In other words, E(x) equals f(g(x)). To determine the marked price, equate E(x) to the amount paid by Izabella. Then, solve the resulting equation for x.
c Compare the total amount paid by each girl and the marked price of the chosen pair of pants.
Solution
a Since Izabella decided to apply the coupon first, the function f will be applied first to the marked price.
g(x)=43x
Substitute
x=f(x)
g(f(x))=43f(x)
Substitute
g(f(x))=I(x)
I(x)=43f(x)
Substitute
f(x)=x−3
I(x)=43(x−3)
I(x)=43(x−3)
Substitute 31 for x and evaluate
Substitute
x=31
I(31)=43(31−3)
SubTerm
Subtract term
I(31)=43(28)
MoveRightFacToNum
ca⋅b=ca⋅b
I(31)=484
CalcQuot
Calculate quotient
I(31)=21
b Emily decided to apply the discounts in the opposite order — that is, she chose to apply the store's sale discount first. For her purchase, the function g will be applied to the marked price.
f(x)=x−3
Substitute
x=g(x)
f(g(x))=g(x)−3
Substitute
f(g(x))=E(x)
E(x)=g(x)−3
Substitute
g(x)=43x
E(x)=43x−3
E(x)=43x−3
Substitute
E(x)=21
21=43x−3
Solve for x
AddEqn
LHS+3=RHS+3
24=43x
MultEqn
LHS⋅4=RHS⋅4
96=3x
DivEqn
LHS/3=RHS/3
32=x
RearrangeEqn
Rearrange equation
x=32
c To determine who saved more money, it would help to organize the results obtained in a table.
| Marked Price | Amount Paid | |
|---|---|---|
| Izabella's Pants | $31 | $21 |
| Emily's Pants | $32 | $21 |
E(31)=43(31)−3=20.25
In contrast, if Emily had applied the discounts as Izabella did, she would have paid $21.75. Regardless of who applied the discounts better, both girls got a big discount on their purchase and went home happy.
Closure
A Special Case of Compositions of Functions
As shown, the composition of functions is not commutative, implying that, in general, f(g(x))=g(f(x)). However, for those functions that do commute, there is a special case. Consider, for example, the following pair of functions.
Next, perform the composition in the opposite order, that is, g(f(x)).
As can be seen, both compositions resulted in the same function.
f(x)=3x−6andg(x)=31x+2
For these two functions, start by finding f(g(x)).
f(x)=3x−6
Substitute g(x) for x and simplify
f(31x+2)=3(31x+2)−6
f(31x+2)=x
g(x)=31x+2
Substitute f(x) for x and simplify
g(3x−6)=31(3x−6)+2
Simplify right-hand side
Distr
Distribute 31
g(3x−6)=31⋅3x−31⋅6+2
a⋅a1=1
g(3x−6)=1⋅x−31⋅6+2
MultByOne
a⋅1=a
g(3x−6)=x−31⋅6+2
MoveRightFacToNum
ca⋅b=ca⋅b
g(3x−6)=x−36+2
CalcQuot
Calculate quotient
g(3x−6)=x−2+2
AddTerms
Add terms
g(3x−6)=x
f(g(x))=g(f(x))=x
The important thing here is not the fact that the functions commute but that the result of the composition is simply x, the original input. In such cases, the functions are said to be inverses of each other. For more information, read about the inverse of a function.