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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

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

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 is a special way to write functions that explicitly shows that $y$ is a function of $x$ — in other words, that $y$ depends on $x.$ Function notation is symbolically expressed as $y=f(x)$ and read ### Extra

Interpreting Function Notation

$y$ equals $f$ of $x.$Equations that are functions can be written using function notation.

$Equation y=-5x+4 Function Notation f(x)=-5x+4 $

Notice that $y$ has been replaced by $f(x).$ In function notation, $x$ represents an element of the domain and $f(x)$ represents the element of the range that corresponds to $x.$ When written in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.
Besides $f,$ other letters such as $g$ or $h$ can be used to name the function. Similarly, letters other than $x$ can name the independent variable.

To interpret an equation given in function notation, it is necessary to understand what both sides of $f(x)=k$ mean. For example, consider the following equation.

$f(3)=12 $

Here, $f(3)$ denotes that the function's input is $x=3$ and that $12$ is the output corresponding to this input.
$f(3)=12⇓The output offwhenx=3is12. $

Now, consider a different scenario. Let $w(t)=200t$ be a function that describes the number of words Kevin reads in $t$ minutes. The following statements are true for this function.
$w(4)andw(t)=900 $

Here, $w(4)$ is the number of words that Kevin reads in $4$ minutes and can be found by evaluating the function for $t=4.$ However, the input is not a particular number in the second statement. In such cases, the statement can be interpreted as a question.
$w(t)=900⇓For which value oftis theoutput equal to900? $

Based on the context, the second statement asks how many minutes it takes Kevin to read $900$ words. To find this value of $t,$ the equation $w(t)=900$ has to be solved for $t.$
Discussion

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.

Method

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 $x=4.$
*expand_more*
*expand_more*

$f(x)=3x+4 $

To evaluate a function for a particular input, there are two steps to follow.
1

Substitute the Input for $x$

Substitute the given input value for every instance of the independent variable in the function. In this case, $4$ is substituted for $x.$

2

Evaluate the Expression

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 $4,$ the output of the function is $16.$

Method

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 $x-$value for which $f(x)=21.$
*expand_more*
*expand_more*

$f(x)=4x−3 $

To find the input that produces a certain output, there are two steps to follow.
1

Substitute the Output for $f(x)$

Start by substituting the given output value for $f(x).$ In this case, it means substituting $21$ for $f(x).$

2

Solve for $x$

Example

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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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.

$C(s)=4.5s−3$

Substitute

$s=10$

$C(10)=4.5(10)−3$

Multiply

Multiply

$C(10)=45−3$

SubTerm

Subtract term

$C(10)=42$

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

$s=16$

Example

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 $V(x)=9(x+1)−3(3−x),$ where $x$ is the length of an inner side of the box — a measure that the girls have to provide.
### Hint

### Solution

Therefore, the volume of the box Emily wants is $12$ cubic inches. Since the price is half the volume, Emily will pay $$6$ for such a box.
Consequently, the output is $21$ when $x=1.75.$ Therefore, the length of the inner side of the box Izabella asked for is $1.75$ inches.

a Emily wants to buy a box where the length of the inner side of the box is $1$ inch. How much will she pay for this box?

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b Izabella paid $$10.50$ for her box. What is the length of the inner side of the box she asked for?

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a Start by finding the volume of the box for the measure Emily chose. To do this, find $V(1).$ The price is half the volume.

b Use the given price to find the volume of Izabella's box. Then, substitute this volume for $V(x)$ in the function rule and solve the resulting equation for $x.$

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 $1$ inch long is given by $V(1).$ Substitute $1$ for $x$ into the function rule and simplify the right-hand side.

$V(x)=9(x+1)−3(3−x)$

▼

Substitute $1$ for $x$ and simplify

Substitute

$x=1$

$V(1)=9(1+1)−3(3−1)$

AddSubTerms

Add and subtract terms

$V(1)=9(2)−3(2)$

Multiply

Multiply

$V(1)=18−6$

SubTerm

Subtract term

$V(1)=12$

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 $$10.50$ for the box.

$2⋅10.50=21in_{3} $

The volume of Izabella's box is $21$ cubic inches. The next step is to find the value of $x$ that produced this volume. To do so, substitute $V(x)$ for $21$ in the function rule and solve the equation for $x.$
$V(x)=9(x+1)−3(3−x)$

Substitute

$V(x)=21$

$21=9(x+1)−3(3−x)$

▼

Solve for $x$

Distr

Distribute $9and-3$

$21=9x+9−9+3x$

AddSubTerms

Add and subtract terms

$21=12x$

DivEqn

$LHS/12=RHS/12$

$1221 =x$

CalcQuot

Calculate quotient

$1.75=x$

RearrangeEqn

Rearrange equation

$x=1.75$

Pop Quiz

Discussion

Two functions $f(x)$ and $g(x)$ can be combined into a sum, a difference, a product, or a quotient through the following formulas.
*composition*. ## Composite Function

$(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 Concept

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 $(f∘g)(x).$
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.
### Extra

Parts of a Composition

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 $g(x).$

$f(x)=2x+1$

▼

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

$h(x)=2x−9$

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.

Explore the parts of a composition by moving the magnifying glass.

Example

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 $g(x)=43 x$ 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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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["21"]}}

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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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["32"]}}

c Which of them saved more money?

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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.

a Since Izabella decided to apply the coupon first, the function $f$ will be applied first to the marked price.

The store's sale discount will be applied to the resulting output. This means that $g$ will be applied to the previous output.

Therefore, the composition of $g$ and $f$ represents the cost that Izabella will pay, based on her decision. This means that $I(x)=g(f(x)).$ To find this composition, in $g(x)=43 x,$ substitute $f(x)$ for $x.$

$g(x)=43 x$

Substitute

$x=f(x)$

$g(f(x))=43 f(x)$

Substitute

$g(f(x))=I(x)$

$I(x)=43 f(x)$

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.

Next, the coupon will be applied. Therefore, the function $f$ will be applied to the previous output.

Consequently, the composition of $f$ and $g$ represents the cost that Emily will pay, based on the order she of discounts she chose — $E(x)=f(g(x)).$ To calculate this composition, evaluate $f(x)$ at $g(x).$

$f(x)=x−3$

Substitute

$x=g(x)$

$f(g(x))=g(x)−3$

Substitute

$f(g(x))=E(x)$

$E(x)=g(x)−3$

Substitute

$g(x)=43 x$

$E(x)=43 x−3$

$E(x)=43 x−3$

Substitute

$E(x)=21$

$21=43 x−3$

▼

Solve for $x$

AddEqn

$LHS+3=RHS+3$

$24=43 x$

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

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.

$f(x)=3x−6andg(x)=31 x+2 $

For these two functions, start by finding $f(g(x)).$
$f(x)=3x−6$

▼

Substitute $g(x)$ for $x$ and simplify

$f(31 x+2)=3(31 x+2)−6$

$f(31 x</$