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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.
Applet to evaluate a function at another function
It can be seen that which suggests that the order does not affect the result. Is it always true? Is it true for and
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.

Concept

Function Notation

Function notation is a special way to write functions that explicitly shows that is a function of — in other words, that depends on Function notation is symbolically expressed as and read equals of Equations that are functions can be written using function notation.
Notice that has been replaced by In function notation, represents an element of the domain and represents the element of the range that corresponds to When written in function notation, the expression that describes how to convert an input into an output — the right-hand side expression — is called the function rule.
f(x)=-5x+4, the right-hand side expression is the function rule

Besides other letters such as or can be used to name the function. Similarly, letters other than can name the independent variable.

Extra

Interpreting Function Notation
To interpret an equation given in function notation, it is necessary to understand what both sides of mean. For example, consider the following equation.
Here, denotes that the function's input is and that is the output corresponding to this input.
Now, consider a different scenario. Let be a function that describes the number of words Kevin reads in minutes. The following statements are true for this function.
Here, is the number of words that Kevin reads in minutes and can be found by evaluating the function for However, the input is not a particular number in the second statement. In such cases, the statement can be interpreted as a question.
Based on the context, the second statement asks how many minutes it takes Kevin to read words. To find this value of the equation has to be solved for
Discussion

Evaluating Functions and Finding a Certain Input

In the same way that an expression can be evaluated at a particular value, functions can also be evaluated at a specific input. Furthermore, it is also possible to determine the input that produces a specific output.

Method

Evaluating a Function

Evaluating a function involves determining the value of the function when its independent variable is set to a specific value. This is done by substituting the given input value for the variable and evaluating the function rule. As an example, consider the value of the following function when
To evaluate a function for a particular input, there are two steps to follow.
1
Substitute the Input for
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Substitute the given input value for every instance of the independent variable in the function. In this case, is substituted for
2
Evaluate the Expression
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Next, evaluate the resulting expression by performing the required operations. The simplified value will be the output for the given input value.
As shown, when the input is the output of the function is
Method

Finding the Input of a Function

Given a function, it is possible to find the input that produces a certain output. This is done by substituting the given output value for the dependent variable and then solving for the independent variable. For the following function, try finding the value for which
To find the input that produces a certain output, there are two steps to follow.
1
Substitute the Output for
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Start by substituting the given output value for In this case, it means substituting for
2
Solve for
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After that, solve the resulting equation for
As shown, is equal to when
When modeling real-life scenarios with functions, in addition to knowing how to evaluate the function and determine a particular input, it is essential to know how to interpret the results obtained.
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 gift card in her purse.

A 3 dollars gift card

The function represents the cost, in dollars, of buying souvenirs with a gift card.

a How much will the girls pay if they buy ten souvenirs?
b How many souvenirs can the girls buy with

Hint

a To find the cost of ten souvenirs, evaluate the given function at Substitute for into the given function. The input represents the number of souvenirs bought and the output represents the cost.
b Find the value of for which the output is Substitute for and solve the resulting equation for

Solution

a Based on the given information, finding how much will the girls pay for buying ten souvenirs is equivalent to finding This can be done by substituting for into the function rule and simplifying the right-hand side.
The statement shows that when the input is the output is In this context, it means that the girls will pay for ten souvenirs.
b In this context, gives the cost of buying souvenirs. Therefore, asking how many souvenirs the girls can buy with is the same as asking for which value of the output of is equal to
To find such value of in the function rule, substitute for and solve the resulting equation for
Solve for
As shown, the output is when In other words, the girls can buy souvenirs with
Example

Evaluating Functions

While walking downtown, Emily and Izabella walked into a craft store and saw a beautiful wooden jewelry box. The craftsman said that the price of the box is half its volume, which is given by the function where is the length of an inner side of the box — a measure that the girls have to provide.
A wooden box.
a Emily wants to buy a box where the length of the inner side of the box is inch. How much will she pay for this box?
b Izabella paid for her box. What is the length of the inner side of the box she asked for?

Hint

a Start by finding the volume of the box for the measure Emily chose. To do this, find The price is half the volume.
b Use the given price to find the volume of Izabella's box. Then, substitute this volume for in the function rule and solve the resulting equation for

Solution

a Since the price depends on the volume, the volume of the box needs to be calculated first. Note that the volume of a box where the inner side is inch long is given by Substitute for into the function rule and simplify the right-hand side.
Substitute for and simplify
Therefore, the volume of the box Emily wants is cubic inches. Since the price is half the volume, Emily will pay for such a box.
b To find the length of the inner side of Izabella's box, the volume of the box needs to be calculated first. If the price is half the volume of the box, then the volume of the box is twice the price paid. Therefore, the volume of the box will be found using the fact that Izabella paid for the box.
The volume of Izabella's box is cubic inches. The next step is to find the value of that produced this volume. To do so, substitute for in the function rule and solve the equation for
Solve for
Consequently, the output is when Therefore, the length of the inner side of the box Izabella asked for is inches.
Pop Quiz

Evaluating Different Functions

Evaluate the given function at the given input.

Random functions that need to be evaluated at random inputs
Discussion

Combining Functions

Two functions and can be combined into a sum, a difference, a product, or a quotient through the following formulas.
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 and is denoted as or
Given four different functions, the composition of two/three of them is found.
Performing the composition of two functions is similar to evaluating one function into the other. For example, let and To find the variable in must be substituted with
Substitute for and evaluate
Note that makes sense only when the outputs of belong to the domain of Also, be aware that the composition of functions is not commutative — that is, in general, This can be checked with the same two functions.
Two machines simulating the composition of two functions. The functions used are f(x)=2x+1 and g(x)=x-5. The order of the composition can be set. Any input between -100 and 100 is accepted.
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 and are different values.

Extra

Parts of a Composition
Explore the parts of a composition by moving the magnifying glass.
Three sets labeled as A, B, and C respectively. Set A is the domain of g and has x inside; Set B is the range of g and also the domain of f. It has g(x) inside it; Set C is the range of f and has f(g(x)) inside; A bendy arrow points from A to B representing the function g; A bendy arrow points from B to C representing the function f; A bendy arrow points from A to C representing the composite function f(g(x)).
Example

Applying Discounts Through Compositions of Functions

Izabella and Emily are shopping for pants. Both girls have a discount coupon for the store. After choosing a pair of pants each, they discovered that the pants are 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.

Two pairs of pants. One of them is marked for 31 dollars. Also, a 3 dollars discount coupon is shown.

The function represents the effect of applying only the coupon to the marked price, while represents the effect of applying only the store's sale discount to the marked price. In both functions, represents the marked price.

a Write a function that represents how much Izabella will pay if she uses the coupon first and then applies the store's sale discount.
Izabella chose a pair of pants whose marked price was How much did she pay for them?
b Write a function that represents how much Emily will pay if she uses the store's discount first and then applies the coupon.
If Emily paid the same amount as Izabella, what was the marked price of the pants Emily bought?
c Which of them saved more money?

Hint

a Applying the coupon first means to apply to the marked price. Then, must be applied to the resulting output. Therefore, is equal to To find how much Izabella paid, evaluate
b This time, apply the store's discount first — that is, start by applying to the marked price. Then, apply to the resulting output. In other words, equals To determine the marked price, equate to the amount paid by Izabella. Then, solve the resulting equation for
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 will be applied first to the marked price.
tagged price: x; applying the coupon: f(x)
The store's sale discount will be applied to the resulting output. This means that will be applied to the previous output.
tagged price: x; applying the coupon: f(x); applying the store's discount: g(f(x))
Therefore, the composition of and represents the cost that Izabella will pay, based on her decision. This means that To find this composition, in substitute for
Izabella chose a pair of pants with a marked price of To find how much she paid for them, evaluate at
Substitute for and evaluate
Consequently, Izabella paid for the pair of pants she chose.
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 will be applied to the marked price.
tagged price: x; applying the store's discount: g(x)
Next, the coupon will be applied. Therefore, the function will be applied to the previous output.
tagged price: x; applying the store's discount: g(x); applying the coupon: f(g(x))
Consequently, the composition of and represents the cost that Emily will pay, based on the order she of discounts she chose — To calculate this composition, evaluate at
Emily paid the same amount as Izabella, which means that Emily paid To determine the original cost of the pair of pants that Emily chose, substitute for and solve the resulting equation for
Solve for
In conclusion, the marked price of the pants Emily chose was
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
Emily's Pants
As shown, both girls paid the same amount of money, but the pair of pants that Emily chose was more expensive. As such, Emily saved more money from the discounts. In fact, if Izabella had applied the discounts as Emily did, she could have paid for the same pair of pants.
In contrast, if Emily had applied the discounts as Izabella did, she would have paid 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, However, for those functions that do commute, there is a special case. Consider, for example, the following pair of functions.
For these two functions, start by finding
Substitute for and simplify
Simplify right-hand side